Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: multiplicity results for stepwise potentials

  • Elisa Ellero1 and

    Affiliated with

    • Fabio Zanolin1Email author

      Affiliated with

      Boundary Value Problems20132013:167

      DOI: 10.1186/1687-2770-2013-167

      Received: 30 November 2012

      Accepted: 11 June 2013

      Published: 15 July 2013

      Abstract

      We prove some multiplicity results for a class of one-dimensional nonlinear Schrödinger-type equations of the form

      x + 2 k x 2 g ( t ) x 3 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equa_HTML.gif

      where k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq1_HTML.gif and the weight g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is a positive stepwise function. Instead of the cubic term, more general nonlinearities can be considered as well.

      MSC: 34C37, 34B15.

      Keywords

      homoclinic solutions heteroclinic solutions multiplicity nonlinear Schrödinger equation stepwise potential topological methods

      Dedication

      Dedicated to Professor Jean Mawhin

      1 Introduction

      This paper deals with the study of homoclinic and heteroclinic solutions for a class of nonlinear second-order differential equations of the form
      x + σ x a ( t ) h ( x ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ1_HTML.gif
      (1.1)
      where σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq3_HTML.gif is a fixed (positive) coefficient and a ( ) : R R 0 + : = ] 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq4_HTML.gif is a bounded weight function. For the nonlinear term, which we split as
      h ( x ) = x f ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equb_HTML.gif

      we suppose that

      (∗) f : R R + : = [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq5_HTML.gif is a locally Lipschitz function which is even, strictly increasing on [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq6_HTML.gif such that f ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq7_HTML.gif and f ( + ) = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq8_HTML.gif.

      Examples of functions f which are suitable for our considerations are, for instance, f ( s ) = s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq9_HTML.gif as in [1], or f ( s ) = s 2 + b s 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq10_HTML.gif ( b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq11_HTML.gif) as in [2]. In both these cases, the function f is even. In Section 3 we briefly describe how this restriction could be avoided.

      Equations of the form (1.1) naturally arise in the search of particular solutions for some classes of nonlinear Schrödinger equations (NLSE) with inhomogeneous nonlinearities. Typical examples are related to the NLSE
      i ħ Ψ t = ħ 2 2 m Δ Ψ + V 1 ( x ) Ψ + V 2 ( x ) | Ψ | p 1 Ψ , x R N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ2_HTML.gif
      (1.2)
      where ħ is the Planck constant, m is the particle’s mass, i is the imaginary unit, Δ is the Laplace operator and p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq12_HTML.gif. In literature, solutions of the form
      Ψ ( t , x ) = u ( x ) exp ( i λ ħ 1 t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equc_HTML.gif
      where λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq13_HTML.gif and u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq14_HTML.gif is a real-valued function, are called stationary waves. Their search leads to the equation
      ħ 2 2 m Δ u + λ u + V 1 ( x ) u V 2 ( x ) | u | p 1 u = 0 , x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ3_HTML.gif
      (1.3)

      (see [3]). Usually, the Planck constant and the mass are omitted in (1.3) after rescaling.

      In many significant models of NLSE, one-dimensional waves are considered. They are studied, for example, in nonlinear optics, in the theory of ocean rogue waves and for Bose-Einstein condensates (just to mention a few cases). For instance, in [1], Belmonte-Beitia and Torres analyzed the one-dimensional equation
      1 2 u + λ u + g ( x ) u 3 = 0 , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ4_HTML.gif
      (1.4)

      which is a particular case of (1.3) with N = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq15_HTML.gif, p = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq16_HTML.gif and V 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq17_HTML.gif. In equation (1.4) the nonlinearity is called inhomogeneous due to the presence of a non-constant weight V 2 ( x ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq18_HTML.gif, which in [1] is assumed to be positive. The sign condition on g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq19_HTML.gif implies that nontrivial bounded solutions of (1.4) can exist only for λ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq20_HTML.gif (see [[1], Theorem 1]). For this reason, we prefer to set λ = k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq21_HTML.gif with k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq22_HTML.gif.

      In order to study equation (1.4) or its variants, we are going to follow a dynamical system approach, hence we choose to treat the independent variable (which in the applications has a spatial connotation) as a time variable, via the substitution x t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq23_HTML.gif. Similarly, for the dependent variable, we make the substitution u x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq24_HTML.gif. In this way, equation (1.4) reads as
      x + 2 k x 2 g ( t ) x 3 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ5_HTML.gif
      (1.5)

      which belongs to the same class of (1.1).

      In spite of the apparent simplicity of equation (1.5), a throughout study of its solutions may be a rather difficult task for a general nonconstant weight function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif. Looking for homoclinic and heteroclinic solutions of (1.5), it will be natural to focus on the behavior of g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif at ±∞. In similar situations, various authors have confined their study to the case in which g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is asymptotically constant [46], or eventually constant [7, 8]. These assumptions are also justified by the analysis of some physical underlying models, in which a layered structure is present. With this respect, see the introduction in [8], where different eventually autonomous cases are listed for related NLSEs arising in nonlinear optics. Examples in which the nonlinear term presents a piecewise constant weight function have been studied also in biological and chemical models. In particular, these situations occur in the theory of wave propagation for reaction-diffusion systems; see, for instance, [911]. In the context of equation (1.5), examples in which g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is a piecewise constant function have been considered as well (see [12]).

      The aim of the present paper is to provide multiplicity results regarding homoclinic and heteroclinic solutions for equation (1.1) under particular assumptions on the weight function a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq25_HTML.gif. Actually, we suppose that a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq25_HTML.gif is an eventually constant piecewise function with only two steps.

      Our approach combines phase-plane analysis with time-mapping estimates. As in [5, 7, 8, 10, 11, 13], the solutions are obtained by connecting the unstable and stable manifolds of the equilibrium points of the asymptotically autonomous equations. Such connections are performed by means of orbit paths of an intermediate equation, which represents the behavior of the system during a suitable interval of transition [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif between the asymptotic states. Multiple connecting solutions arise when such interval length is sufficiently large. Lower estimates for t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq27_HTML.gif will be provided in terms of time mappings, which can be expressed by Abelian integrals. This kind of approach is also reminiscent of some topological methods for the study of Sturm-Liouville boundary value problems. Indeed, a solution that satisfies the Sturm-Liouville boundary conditions can be interpreted as a trajectory in the phase-plane that connects two lines (see [1417]). Generalized Sturm-Liouville solutions, which connect the graphs of two functions or given planar continua, have been considered as well (see [1820]). Often these problems can be settled in the framework of the theory of ODEs with nonlinear boundary conditions (see [21, 22]).

      In order to make our approach more transparent, we are going to perform our analysis for equation (1.5). This choice is motivated by the sake of avoiding unnecessary technicalities. Our arguments can be modified in a straightforward manner in the case of more general equation (1.1), with f satisfying (∗) (see Section 3). Homoclinic and heteroclinic orbits can be interpreted as solutions for some boundary value problems on unbounded intervals. In the last section we also outline possible applications of our approach to boundary value problems on a compact interval (like the Sturm-Liouville one).

      Besides his manifold achievements in different areas of mathematics, Professor Jean Mawhin is one of the pioneers in the study of topological methods for nonlinear boundary value problems. It is a pleasure and an honor to have the possibility to dedicate our work to his important contributions in this area.

      2 Homoclinic and heteroclinic solutions: multiplicity results

      2.1 General setting

      We consider the second-order nonlinear differential equation
      x + 2 k x 2 g ( t ) x 3 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ6_HTML.gif
      (2.1)
      and its equivalent system in the phase-plane
      { x = y , y = 2 k x + 2 g ( t ) x 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ7_HTML.gif
      (2.2)

      where k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq1_HTML.gif is a fixed coefficient and g : R R 0 + : = ] 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq28_HTML.gif is a bounded measurable function. Solutions of (2.1) are meant in the generalized (i.e., Carathéodory) sense (see [23]). Actually, in our results a step function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif with only two jumps is considered and the solutions are piecewise smooth.

      In the particular case of a constant coefficient g ( t ) μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq29_HTML.gif, an elementary analysis of the system
      { x = y , y = 2 k x + 2 μ x 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ8_HTML.gif
      (2.3)
      shows that the associated phase portrait presents a center O = ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq30_HTML.gif and two saddle points P ± : = ( ± ( k / μ ) 1 2 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq31_HTML.gif. These saddles are connected by two heteroclinic solutions (one connecting P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif in the upper half-plane and a symmetric one from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif in the lower half-plane). The heteroclinic orbits are described by
      y 2 = μ ( x 2 ( k / μ ) ) 2 for  ( k / μ ) 1 2 < x < ( k / μ ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ9_HTML.gif
      (2.4)

      In [1] Belmonte-Beitia and Torres supposed that the weight function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is even and T-periodic and proved the existence of a heteroclinic solution connecting two periodic stationary states. Such a result generalizes to the case of periodic coefficients the situation described above for the autonomous system.

      As already observed in the introduction, equation (2.1) has been already studied by various authors for its relevance in many applicative models. The aim of the present paper is to provide multiplicity results for heteroclinic and also homoclinic solutions of equation (2.1). A possible way to obtain this goal is to assume that the weight function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif has a different behavior at infinity and in some intermediate time interval. In this setting, a first step consists of analyzing the situation in which g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is a stepwise function assuming only two values. We believe that the study of such a simplified case may lead to more general considerations, in which, for instance, g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is eventually periodic. In this way, multiplicity results for Belmonte-Beitia and Torres’s model can be obtained (see Section 3 for a brief overview concerning possible applications of our approach). Equation (2.1) with a periodic stepwise coefficient g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif has recently been studied in [24] in the context of chaotic-like dynamics.

      As a tool, in our paper, we use the Poincaré map associated to system (2.2). Given a fixed time interval [ t 0 , t 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq34_HTML.gif and a point z R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq35_HTML.gif, we indicate by ζ ( , t 0 , z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq36_HTML.gif the solution ζ ( t ) = ( x ( t ) , y ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq37_HTML.gif of (2.2) satisfying the initial condition ζ ( t 0 ) = z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq38_HTML.gif. The Poincaré map on [ t 0 , t 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq34_HTML.gif is defined as
      Φ = Φ t 0 t 1 : z ζ ( t 1 , t 0 , z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equd_HTML.gif

      From the fundamental theory of differential equations it follows that the domain D Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq39_HTML.gif of Φ is an open subset of the plane and Φ is an orientation preserving homeomorphism of D Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq39_HTML.gif onto its image Φ ( D Φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq40_HTML.gif.

      2.2 Analysis of the equation

      Let us consider system (2.2) for the stepwise weight function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif, defined as
      g ( t ) = { μ 1 for  t t  or  t t + , μ 0 for  t < t < t + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ10_HTML.gif
      (2.5)
      where
      t < t + and 0 < μ 0 < μ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Eque_HTML.gif
      System (2.2) can be seen as the superposition of the autonomous systems
      { x = y , y = 2 k x + 2 μ 1 x 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ11_HTML.gif
      (2.6)
      and
      { x = y , y = 2 k x + 2 μ 0 x 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ12_HTML.gif
      (2.7)
      The first one describes the asymptotic behavior of the equation. We are going to use the orbits of the second system for connecting unstable/stable manifolds of (2.6) during the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif. Figure 1 shows the superposition of the phase portraits of the two systems.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig1_HTML.jpg
      Figure 1

      In the present figure we have considered the superposition of the phase portraits of systems (2.6) (in darker color) and (2.7) (in lighter color) with k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq41_HTML.gif, μ 1 = 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq42_HTML.gif, μ 0 = 1 / 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq43_HTML.gif. For graphical reasons, a slightly different x- and y-scaling has been used.

      First of all, we briefly analyze the structure of system (2.6). Its equilibrium points are the origin (which is a center) and two saddle points P ± = ( ± ξ 1 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq44_HTML.gif, where
      ξ 1 : = ( k / μ 1 ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equf_HTML.gif
      The stable and unstable manifolds of P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq45_HTML.gif are illustrated in Figure 2. The sets I ( P ) L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq46_HTML.gif and O ( P ) L ( P , P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq47_HTML.gif are, respectively, the stable and the unstable manifolds of  P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif. Symmetrically, I ( P + ) L ( P , P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq49_HTML.gif and O ( P + ) L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq50_HTML.gif are the stable and the unstable manifolds of P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig2_HTML.jpg
      Figure 2

      The stable and unstable manifolds of P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq52_HTML.gifand P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq53_HTML.gif. The arrows show the direction of the flow along the orbits.

      In the case of the cubic nonlinearity, we find
      I ( P ) = { ( x , y ) : y = | μ 1 x 2 k | / μ 1 , x < ( k / μ 1 ) 1 2 } , O ( P ) = { ( x , y ) : y = | μ 1 x 2 k | / μ 1 , x < ( k / μ 1 ) 1 2 } , I ( P + ) = { ( x , y ) : y = | μ 1 x 2 k | / μ 1 , x > ( k / μ 1 ) 1 2 } , O ( P + ) = { ( x , y ) : y = | μ 1 x 2 k | / μ 1 , x > ( k / μ 1 ) 1 2 } , L ( P , P + ) = { ( x , y ) : y = | μ 1 x 2 k | / μ 1 , ( k / μ 1 ) 1 2 < x < ( k / μ 1 ) 1 2 } , L ( P + , P ) = { ( x , y ) : y = | μ 1 x 2 k | / μ 1 , ( k / μ 1 ) 1 2 < x < ( k / μ 1 ) 1 2 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equg_HTML.gif

      The solutions of system (2.2) with g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif defined as in (2.5) will be obtained by connecting suitably chosen parts of stable and unstable manifolds of (2.6) with trajectories of (2.7). More in detail, we proceed as follows.

      Due to the Hamiltonian nature of the equation under consideration, we can consider the ‘energy’ level lines of system (2.7), given by
      E ( x , y ) : = y 2 + 2 k x 2 μ 0 x 4 = c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ13_HTML.gif
      (2.8)
      We also set
      Γ c : = { ( x , y ) : ξ 0 x ξ 0 , E ( x , y ) = c } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equh_HTML.gif
      where
      ξ 0 : = ( k / μ 0 ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equi_HTML.gif
      The curve Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif is the part of the line at level c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq55_HTML.gif in the strip [ ξ 0 , ξ 0 ] × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq56_HTML.gif. For every c such that
      0 < c < 0 : = k 2 μ 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equj_HTML.gif
      the set Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif is a periodic orbit of (2.7) run in the clockwise sense, which intersects the y-axis at the points ( 0 , ± c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq57_HTML.gif. The (minimal) period τ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq58_HTML.gif of Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif can be expressed by the following time-mapping formula:
      τ c = 2 s 0 s 1 d x c 2 F ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ14_HTML.gif
      (2.9)
      where F ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq59_HTML.gif is the potential associated to the equation and s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq60_HTML.gif, s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq61_HTML.gif are such that
      F ( s 0 ) = F ( s 1 ) = c / 2 with  ξ 0 < s 0 < 0 < s 1 < ξ 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equk_HTML.gif
      In our case,
      F ( x ) = k x 2 μ 0 2 x 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equl_HTML.gif
      and by symmetry s 0 = s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq62_HTML.gif. More in detail, we can express the period τ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq58_HTML.gif by means of an elliptic integral in the following way:
      τ c = 4 μ 0 0 s 1 d x ( a 2 x 2 ) ( b 2 x 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ15_HTML.gif
      (2.10)
      for a : = s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq63_HTML.gif, b : = ( 2 k 2 μ 0 2 s 1 2 ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq64_HTML.gif and 0 < s 1 < ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq65_HTML.gif, 0 < c = 2 k s 1 2 μ 0 s 1 4 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq66_HTML.gif. Notice that 0 < a < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq67_HTML.gif in (2.10). Therefore
      τ c = 4 b μ 0 sn 1 ( 1 , a / b ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equm_HTML.gif

      where sn ( , k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq68_HTML.gif is the Jacobi elliptic sine function of modulus k : = a / b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq69_HTML.gif (see [25, 26]).

      The level line Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif contains the saddle points Q ± : = ( ± ξ 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq71_HTML.gif of system (2.7) and their heteroclinic connections L ( Q , Q + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq72_HTML.gif in the upper half-plane and L ( Q + , Q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq73_HTML.gif in the lower half-plane. The energy level
      1 : = k 2 μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equn_HTML.gif
      corresponds to the closed curve Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif, which is tangent to L ( P , P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq75_HTML.gif and L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq76_HTML.gif. Moreover, for
      : = k 2 μ 1 ( 2 μ 0 μ 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equo_HTML.gif

      the energy level line Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif contains the saddle points P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq45_HTML.gif of system (2.6).

      Using the parameters 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq78_HTML.gif, 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq80_HTML.gif, we define the regions
      E : = { ( x , y ) : ξ 0 x ξ 0 , 1 E ( x , y ) } , F : = { ( x , y ) : ξ 0 x ξ 0 , < E ( x , y ) < 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equp_HTML.gif
      Both ℰ and ℱ are invariant sets for (2.7); they are filled by periodic orbits. In the sequel, the orbits in the region ℰ will be called internal, while those in ℱ will be called external. The choice of these names is made in order to distinguish the trajectories with respect to  Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif, which contains the saddle points P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq45_HTML.gif of system (2.6). (See Figure 3.)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig3_HTML.jpg
      Figure 3

      The energy level lines Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq81_HTML.gif, Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq82_HTML.gifand Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq83_HTML.giffor system (2.7) superimposed over the stable and unstable manifolds of P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq84_HTML.giffor system (2.7). The colored regions are the invariant sets ℰ (the darker one) and ℱ (lighter). The figure has been plotted with the parameters k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq86_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq87_HTML.gif. For graphical reasons, a slightly different x- and y-scaling has been used.

      In order to obtain homoclinic or heteroclinic solutions for system (2.2) with g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif defined as in (2.5), we connect the unstable manifold of the point A { P , P + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq88_HTML.gif to the stable manifold of B { P , P + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq89_HTML.gif via an orbit path of Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif for c [ 1 , 0 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq90_HTML.gif. Actually, this is the only way to get the desired solutions. If c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq91_HTML.gif, such connection will follow a trajectory ( x ( t ) , y ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq92_HTML.gif with ξ 0 x ( t ) ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq93_HTML.gif, so it will lie in the invariant annular region ℰ bounded by Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif and Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif. On the contrary, for c > http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq94_HTML.gif the connection lies in the set ℱ between Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif and Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif.

      To provide more explicit details, we consider the case A = P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq95_HTML.gif. A solution of (2.2), which is homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif, can be produced in two ways. One consists in connecting O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif to I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq97_HTML.gif, using an orbit path of Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif for c ] , 0 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq98_HTML.gif. Another possibility is given by connecting a point of L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq76_HTML.gif to a point of L ( P , P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq75_HTML.gif via an orbit path of Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif for c [ 1 , [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq99_HTML.gif. For c = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq100_HTML.gif, we can obtain our solution if and only if t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq101_HTML.gif is an integer multiple of the fundamental period τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq102_HTML.gif of the orbit Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif. Indeed, such a solution will be constantly equal to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif for t ] t , t + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq103_HTML.gif, and for t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq104_HTML.gif it coincides with the periodic solution of (2.7) such that x ( t ) = ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq105_HTML.gif and y ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq106_HTML.gif. This solution makes ( t + t ) / τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq107_HTML.gif turns around the origin for t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq108_HTML.gif.

      Similar considerations can be developed with respect to heteroclinic solutions. For instance, a heteroclinic orbit from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif can be obtained as follows: by connecting a point of O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif with one of I ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq109_HTML.gif using an orbit path of Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif for c ] , 0 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq110_HTML.gif, or by a connection of two different points of L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq76_HTML.gif via an orbit path of Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif for c [ 1 , [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq111_HTML.gif. For c = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq112_HTML.gif, we can obtain our solution if and only if t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq101_HTML.gif is an odd multiple of τ / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq113_HTML.gif, where τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq102_HTML.gif is the period of the orbit Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif. Indeed, such a solution will be constantly equal to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif for t ( , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq114_HTML.gif and constantly equal to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif for t [ t + , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq115_HTML.gif. Moreover, it coincides for t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq104_HTML.gif with the solution of (2.7) such that x ( t ) = ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq105_HTML.gif and y ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq106_HTML.gif. This solution makes 2 ( t + t ) / τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq116_HTML.gif half-turns around the origin for t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq104_HTML.gif.

      Figure 4, although not exhaustive of all the conceivable cases, summarizes several different possibilities. Indeed, a trajectory homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif can be obtained as follows: move by system (2.6) along O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to the intersection point of O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif with a closed curve external to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif, namely δ (for t ( , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq117_HTML.gif). This point is put in evidence with a small black circle. Next, follow δ (by system (2.7)) in order to reach the intersection of δ with I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq97_HTML.gif (for t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq104_HTML.gif). Such an intersection point is indicated by a grey square. Finally, switch to system (2.6) and move toward P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif along I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq97_HTML.gif (for t [ t + , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq118_HTML.gif). With a similar procedure, we can obtain a heteroclinic trajectory from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif. In fact, we can move by system (2.6) along O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to the intersection point of O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif with a closed curve external to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif, namely γ (for t ( , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq117_HTML.gif). Next, follow γ (by system (2.7)) to the intersection with I ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq109_HTML.gif (for t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq104_HTML.gif). Finally, switch to system (2.6) and move toward P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif along I ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq109_HTML.gif (for t [ t + , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq115_HTML.gif).
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig4_HTML.jpg
      Figure 4

      In the present figure we have considered the superposition of (2.6) and (2.7) for k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq41_HTML.gif, μ 1 = 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq42_HTML.gif, μ 0 = 1 / 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq43_HTML.gif. The closed curves α, β, γ, δ represent four different level lines of system (2.7): the curves α and β in ℰ correspond to a level c ] 1 , [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq119_HTML.gif, while γ and δ in ℱ refer to c ] , 0 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq110_HTML.gif. In order to describe a connection from an unstable manifold to a stable one of system (2.6) via an orbit path of (2.7), we have marked with a black circle possible starting points and with a grey square some available end points on the same level lines.

      In general, the connections through the trajectories of system (2.7) will either involve only an arc of the closed curve Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif, or they will require to perform a certain number of winds around the origin, depending on the available time. If the length of the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif is small, only the first case is possible. However, if t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq101_HTML.gif is large enough, more choices arise. For instance, looking at the orbit δ in Figure 4, we could make a certain number of loops from the black circle before reaching the grey square on the same orbit. Similar considerations apply to the heteroclinic trajectory described above with reference to γ.

      Until now we have described the ‘external connections’, namely those which lie in the region ℱ. Further possibilities appear if we consider ‘internal connections’ by means of orbit paths contained in ℰ. For example, if we look at the orbit β in Figure 4, we can obtain trajectories homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif by choosing one of the two black circles as a starting point from L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq76_HTML.gif and one of the two grey squares as an end point on L ( P , P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq75_HTML.gif. Loops along β will be permitted if the time is sufficiently large. The construction of heteroclinic orbits from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif via the line α follows a similar procedure.

      2.3 Study of the Poincaré map

      Another possible point of view to describe the previous construction of homoclinic and heteroclinic solutions consists in considering the Poincaré map associated to system (2.7) for the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif. Due to the autonomous nature of this system, we have that Φ t t + = Φ 0 t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq120_HTML.gif, denoted simply by Φ when no confusion may occur. Moreover, the region
      W : = { ( x , y ) : ξ 0 x ξ 0 , E ( x , y ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equq_HTML.gif
      is a compact invariant set contained in the domain of Φ. Let us denote by Y : = O ( P + ) ¯ F ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq121_HTML.gif the part of the unstable manifold O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif between P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif and Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif, including the extreme points. Hence the external connections can be precisely described by looking at the intersection points of Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif with I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq97_HTML.gif (for the homoclinics) and with I ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq109_HTML.gif (for the heteroclinics), as illustrated in Figure 5.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig5_HTML.jpg
      Figure 5

      In the present figure we show the transformation of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq123_HTML.gif under the Poincaré map Φ for t + t = 5.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq124_HTML.gif. The parameters used are k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq125_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq126_HTML.gif. For graphical reasons a slightly different x- and y-scaling has been used.

      With reference to Figure 5, we observe that the intersection point of O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif with Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif (indicated by a small circle) is moved by Φ along Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif to a point very close to Q + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq127_HTML.gif. On the other hand, during the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif, the point P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif makes a little more than a complete turn around the origin; the final position Φ ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq128_HTML.gif is indicated by a cross. The arc Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif is a spiral-like curve with one end near Q + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq129_HTML.gif and the other on Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif. For the time length t + t = 5.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq130_HTML.gif, considered in Figure 5, there are precisely two intersections of Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif with I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq97_HTML.gif, corresponding to external homoclinic solutions. Meanwhile, we have also an external heteroclinic solution due to the intersection of Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif with I ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq109_HTML.gif. All these three intersection points have been indicated by a grey square (following the same convention used in Figure 4).

      In order to describe further these solutions, let us consider their energy at the time t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq131_HTML.gif that we denote by k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq132_HTML.gif, k 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq133_HTML.gif, k 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq134_HTML.gif, with < k 3 < k 2 < k 1 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq135_HTML.gif. Let us define the points H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq136_HTML.gif as { H i } = O ( P + ) Γ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq137_HTML.gif for i = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq138_HTML.gif. The solution ζ ( t ) = ( x ( t ) , y ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq37_HTML.gif of (2.2) with ζ ( t ) = H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq139_HTML.gif or ζ ( t ) = H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq140_HTML.gif has the following behavior: ζ ( t ) P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq141_HTML.gif for t ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq142_HTML.gif, with x ( t ) > ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq143_HTML.gif and convex for t ] t , t + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq144_HTML.gif. Moreover, x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq145_HTML.gif is increasing on ( , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq146_HTML.gif and decreasing on [ t + , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq147_HTML.gif. On the interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif, if i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq148_HTML.gif, the solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq145_HTML.gif is concave, while if i = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq149_HTML.gif, the solution has two maxima and one (negative) minimum separated by two simple zeros. The solution of system (2.2) with ζ ( t ) = H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq150_HTML.gif is such that ζ ( t ) P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq141_HTML.gif for t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq151_HTML.gif with x ( t ) > ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq152_HTML.gif and ζ ( t ) P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq153_HTML.gif for t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq154_HTML.gif with x ( t ) < ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq155_HTML.gif; x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq145_HTML.gif is convex for t ] t , t + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq156_HTML.gif. Moreover, the solution has one maximum and one minimum separated by one simple zero.

      Even if the time interval length t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq157_HTML.gif is small, we always have at least one intersection of Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif with I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq97_HTML.gif and thus a homoclinic solution. On the other hand, if t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq158_HTML.gif grows, the spiral curve Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif will wind more times around the origin, hence more homoclinic/heteroclinic solutions will appear.

      If we look for the internal connections (made by orbit paths lying in the region ℰ), we proceed as follows.

      Let us set L : = L ( P + , P ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq159_HTML.gif and denote by L : = L ( P , P + ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq160_HTML.gif its symmetric part with respect to the x-axis. Figure 6 illustrates the involved geometry from the point of view of the Poincaré map. We observe that all the points of ℒ are contained in ℰ, hence they are periodic points (of different periods). As before, during the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif, the point P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif moves clockwise around the origin along Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq161_HTML.gif; the final position Φ ( P + ) Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq162_HTML.gif is indicated by a cross. Symmetrically, the point P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif performs exactly the same angle around the origin, reaching the final position Φ ( P ) Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq163_HTML.gif, indicated by a small circle. All the other points of L ( P + , P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq76_HTML.gif move, under the action of Φ, on the energy level lines Γ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq54_HTML.gif with 1 c < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq164_HTML.gif. The arc Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif is a curve contained in ℰ, connecting the two antipodal points Φ ( P ± ) Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq166_HTML.gif and leaning on one point of Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif. This tangent point is the image through Φ of the point R : = ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq167_HTML.gif. Such a property of Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif implies that Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif intersects both ℒ and L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq168_HTML.gif (at different points). These two points have been indicated by a grey square in the figure. Accordingly, we always find at least an internal homoclinic solution and an internal heteroclinic one.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig6_HTML.jpg
      Figure 6

      In the present figure we show the transformation of L : = L ( P + , P ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq169_HTML.gifunder the Poincaré map Φ for t + t = 5.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq170_HTML.gif, the same time considered in Figure5. The parameters used are k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq125_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq171_HTML.gif, as before. The point indicated by a cross is precisely the same point Φ ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq128_HTML.gif of Figure 5. Notice that for this time interval the point makes a little more than a complete turn around the origin. For graphical reasons, a slightly different x- and y-scaling has been used.

      Summarizing the above information, we conclude that, for every time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif, we have at least two solutions which are homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif (one external and one internal) and one internal heteroclinic solution from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif.

      For the external connections, if the time interval length t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq172_HTML.gif grows, the situation can be summarized as follows. The number of external homoclinic/heteroclinic solutions increases, depending on the number of winds around the origin of the curve Φ ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq122_HTML.gif. Indeed, the end point of such a curve on Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif cannot go beyond Q + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq127_HTML.gif, while the other end point Φ ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq128_HTML.gif is free to move on the periodic orbit Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq161_HTML.gif. If we denote by η ext http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq173_HTML.gif and ν ext http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq174_HTML.gif the number of external homoclinic and heteroclinic solutions respectively, it holds that 0 η ext ν ext 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq175_HTML.gif. From the above discussion, we can conclude that if, for some nonnegative integer n, we have t + t > n τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq176_HTML.gif (where τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq102_HTML.gif is the period of Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif), then η ext n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq177_HTML.gif. An analogous lower bound can be provided for ν ext http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq178_HTML.gif. For a formal proof, see Theorem 2.1.

      When t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq157_HTML.gif grows, the situation for the internal homoclinic/heteroclinic solutions is more intriguing, as illustrated in Figure 7 and Figure 8. A twist effect depending on the different periods of Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif and Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif produces a double spiral-like curve Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif when t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq158_HTML.gif is sufficiently large.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig7_HTML.jpg
      Figure 7

      In the present figure we show the transformation of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq179_HTML.gif under the Poincaré map Φ for t + t = 50 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq180_HTML.gif. The parameters used are k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq125_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq126_HTML.gif, as before. For graphical reasons, a slightly different x- and y-scaling has been used.

      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig8_HTML.jpg
      Figure 8

      The present figure shows the evolution of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq179_HTML.gif under the Poincaré map Φ for time intervals of different lengths. The parameters used are k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq125_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq171_HTML.gif, as before, while for t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq157_HTML.gif, we have considered the cases 35, 60, 100, 150 (from the left to the right). For graphical reasons, a slightly different x- and y-scaling has been used.

      The fact that Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif looks like a double spiral depends on the different velocities of the points P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq45_HTML.gif and R = ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq181_HTML.gif (which is the intersection of ℒ with the negative y-axis). In fact, since τ 1 < τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq182_HTML.gif, the points on Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif move faster than those of Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif. By construction, P ± Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq183_HTML.gif, while R Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq184_HTML.gif. Therefore, when the time gap t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq27_HTML.gif is sufficiently large, the number of turns of Φ ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq185_HTML.gif around the origin will exceed the number of turns of Φ ( P ± ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq186_HTML.gif. Accordingly, the image through Φ of the right part of ℒ (the sub-arc of ℒ connecting P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to R) is a spiral that winds a certain number of times in the clockwise sense around the origin. Indeed, it connects the slower point Φ ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq128_HTML.gif to the faster one Φ ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq185_HTML.gif inside the region ℰ, as illustrated in Figure 9. The number of half-turns of this spiral depends on a ‘rotational gap’ that, for our purposes, it will be convenient to define as
      w ( t + t ) : = 2 ( t + t ) τ 1 2 ( t + t ) τ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ16_HTML.gif
      (2.11)
      Similarly, the image through Φ of the left part of ℒ (the sub-arc of ℒ connecting R to  P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif) is a spiral-like curve winding a certain number of times in the counterclockwise sense around the origin. Indeed, it connects the points Φ ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq185_HTML.gif and Φ ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq187_HTML.gif inside the region ℰ (the first point moves at a faster speed than the second one). Again, the number of half-turns of this second curve will depend on w ( t + t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq188_HTML.gif. In the end, gluing together the two spiral-like curves, we conclude that, if t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq101_HTML.gif is sufficiently large, the arc Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif will appear like a double spiral with a central ‘hook’. To better describe the structure of Φ ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq165_HTML.gif, we should observe that the ‘tip of the hook’ is not Φ ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq185_HTML.gif, but it is the image through Φ of a point (close to R) on the left part of L { R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq189_HTML.gif (we owe this remark to the referee). For a formal proof based on the argument outlined above, see Theorem 2.2.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig9_HTML.jpg
      Figure 9

      The present figure explains the twist effect due to the different periods of Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq82_HTML.gifand Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq83_HTML.gif. We have considered the evolution of the part of ℒ with x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq190_HTML.gif connecting P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to the point R for two different time gaps: t + t = 5.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq130_HTML.gif and t + t = 50 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq191_HTML.gif. The corresponding two images Φ ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq128_HTML.gif are indicated by a cross, while the images Φ ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq185_HTML.gif are marked by a black dot. The parameters used are k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq125_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq171_HTML.gif, as before. Notice that the two images Φ ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq128_HTML.gif, although close to each other in the figure, have made a different number of turns around the origin. For graphical reasons, a slightly different x- and y-scaling has been used.

      In preparation for this theorem, we introduce the following notation. Given a positive real number S and an energy level [ 1 , ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq192_HTML.gif, we denote by n ̲ ( S , ) : = 2 S τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq193_HTML.gif the lower integer part of S / ( τ / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq194_HTML.gif, and by n ¯ ( S , ) : = 2 S τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq195_HTML.gif the upper integer part of S / ( τ / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq194_HTML.gif. By definition, during a time interval of length S, a point on the orbit Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq196_HTML.gif makes more that n ¯ ( S , ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq197_HTML.gif half-turns around the origin, but less that n ̲ ( S , ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq198_HTML.gif half-turns. Thus the rotational gap defined above can be written as w ( T ) = n ¯ ( T , 1 ) n ̲ ( T , ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq199_HTML.gif for T = t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq200_HTML.gif. By definition, in the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq201_HTML.gif, the smallest integer is n ̲ ( T , ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq202_HTML.gif, while the largest integer is n ¯ ( T , ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq203_HTML.gif. Hence, the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq204_HTML.gif contains w ( T ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq205_HTML.gif positive integers.

      2.4 Conclusion

      After the preliminary study of the previous subsections, we are now in a position to express our results for the equation
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equr_HTML.gif
      as statements with a formal proof. Recall that the weight function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif is defined as follows:
      g ( t ) = { μ 1 for  t t  or  t t + , μ 0 for  t < t < t + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equs_HTML.gif
      where
      t < t + and 0 < μ 0 < μ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equt_HTML.gif
      We also set T : = t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq206_HTML.gif and consider the Poincaré map Φ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq207_HTML.gif associated to the system
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equu_HTML.gif

      In order to make the exposition more clear, we have decided to consider separately the cases of external and internal connections.

      Theorem 2.1 (External connections)

      Under the above assumptions, the following results hold for solutions which satisfy the energy condition
      x ( t ) 2 + 2 k x ( t ) 2 μ 0 x ( t ) 4 ] , 0 [ , t R : http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equv_HTML.gif
      • there always exist a (positive) solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifand a (negative) solution homoclinic to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif;

      • if t + t > τ / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq208_HTML.gif, there always exist a heteroclinic solution from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifto P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifand a heteroclinic solution from P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifto P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif, both with exactly one zero;

      • if t + t > n τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq209_HTML.gif (for some integer n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq210_HTML.gif) then, for each integer j with 1 j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq211_HTML.gif, there exist at least one solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifand one solution homoclinic to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif, both with exactly 2j zeros;

      • if t + t > ( n + 1 ) τ / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq212_HTML.gif (for some integer n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq210_HTML.gif) then, for each integer j with 1 j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq211_HTML.gif, there exist at least one heteroclinic solution from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifto P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifand one from P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifto P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif, both with exactly 2 j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq213_HTML.gifzeros.

      Proof As a first step, we focus our attention on the search of solutions homoclinic to the point P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif. The case of homoclinics to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif is analogous, thus it will be omitted.

      Recalling that T = t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq214_HTML.gif, our goal is to find points on O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif which are moved by the Poincaré map Φ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq207_HTML.gif on I ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq215_HTML.gif.

      We introduce in the phase-plane for (2.7), a system of polar coordinates ( θ , ρ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq216_HTML.gif, with center in the origin. The initial points in the arc http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq217_HTML.gif (which is the closure of the intersection of O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif with the external region ℱ) are parameterized as γ ( s ) : = ( ρ ( s ) cos θ ( s ) , ρ ( s ) sin θ ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq218_HTML.gif, for s [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq219_HTML.gif, with γ ( 0 ) = P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq220_HTML.gif, γ ( 1 ) Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq221_HTML.gif and E ( γ ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq222_HTML.gif strictly increasing with s. The target set is the symmetric of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq217_HTML.gif with respect to the x-axis. It will be denoted by Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq223_HTML.gif and parameterized by reversing the angle θ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq224_HTML.gif.

      Using the same polar coordinates to represent the solutions of (2.7), we can express the final points Φ 0 T ( γ ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq225_HTML.gif as
      Φ 0 T ( γ ( s ) ) = ( ρ ( T , s ) cos θ ( T , s ) , ρ ( T , s ) sin θ ( T , s ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equw_HTML.gif
      with
      ρ ( 0 , s ) = ρ ( s ) , θ ( 0 , s ) = θ ( s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equx_HTML.gif
      We notice that
      ( x ( t ) , y ( t ) ) = ( ρ ( t , s ) cos θ ( t , s ) , ρ ( t , s ) sin θ ( t , s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equy_HTML.gif
      is the solution of (2.7) (at the time t), which departed from the point γ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq226_HTML.gif at the time t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq227_HTML.gif. Observe that, for every s [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq219_HTML.gif, the map
      t θ ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equz_HTML.gif

      is strictly decreasing on [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq6_HTML.gif (this is an equivalent way to express the fact that the solutions turn around the origin in the clockwise sense).

      With these positions, we obtain an external solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif if and only if there exists s ˆ ] 0 , 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq228_HTML.gif such that Φ 0 T ( γ ( s ˆ ) ) Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq229_HTML.gif. This happens if and only if
      ρ ( T , s ˆ ) = ρ ( s ˆ ) and θ ( T , s ˆ ) = θ ( s ˆ ) 2 j π , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ17_HTML.gif
      (2.12)

      for some nonnegative integer j. In this case, if we denote by x ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq230_HTML.gif the corresponding homoclinic solution of (2.1) such that ( x ˆ ( t ) , x ˆ ( t ) ) = γ ( s ˆ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq231_HTML.gif, we have that E ( x ˆ ( t ) , x ˆ ( t ) ) ] , 0 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq232_HTML.gif for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq233_HTML.gif. Moreover, if j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq234_HTML.gif, x ˆ ( t ) > ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq235_HTML.gif for all t [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq236_HTML.gif, while x ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq237_HTML.gif has precisely 2j zeros in ] t , t + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq238_HTML.gif if j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq239_HTML.gif.

      Using the fact that the energy level lines of (2.7) in the region W { O } = { ( x , y ) : ξ 0 x ξ 0 , 0 < E ( x , y ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq240_HTML.gif are strictly star-shaped with respect to the origin and symmetric with respect to the x-axis, we find that (2.12) holds (for some nonnegative integer j) if and only if
      θ ( T , s ˆ ) = θ ( s ˆ ) 2 j π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ18_HTML.gif
      (2.13)

      is satisfied (for the same j).

      By virtue of (2.13), we can refer only to the angular coordinates, hence we will obtain solutions as follows. Since the point Φ 0 T ( γ ( 1 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq241_HTML.gif lies on Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq70_HTML.gif (which is the trajectory of a heteroclinic solution of (2.7)), it can never reach the point Q + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq127_HTML.gif. Accordingly, the angle θ ( T , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq242_HTML.gif satisfies
      θ ( T , 1 ) > 0 , T 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equaa_HTML.gif
      On the other hand, as γ ( 0 ) = P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq220_HTML.gif, we have that θ ( 0 , 0 ) = θ ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq243_HTML.gif, therefore
      θ ( T , 0 ) < 0 , T > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equab_HTML.gif

      As a consequence, θ ( T , 0 ) + θ ( 0 ) < 0 < θ ( T , 1 ) + θ ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq244_HTML.gif and the intermediate value theorem ensures the existence of an s ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq245_HTML.gif such that θ ( T , s ˆ ) = θ ( s ˆ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq246_HTML.gif. In this way we have found a positive homoclinic solution to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif, independently of the length T of the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif.

      Suppose now that T > n τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq247_HTML.gif for some positive integer n. In this case,
      θ ( T , 0 ) < 2 n π , T > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equac_HTML.gif

      As a consequence, for every integer j with 1 j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq248_HTML.gif, we have θ ( T , 0 ) + θ ( 0 ) < 2 j π < 0 < θ ( T , 1 ) + θ ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq249_HTML.gif, and again the intermediate value theorem ensures the existence of an s ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq245_HTML.gif such that θ ( T , s ˆ ) = θ ( s ˆ ) 2 j π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq250_HTML.gif. This ends the proof for homoclinic solutions.

      For the search of heteroclinic solutions from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif, we follow a similar procedure, choosing as a target the set Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq251_HTML.gif, which is the symmetric of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq217_HTML.gif with respect to the y-axis (here we exploit the oddness of the nonlinear term x 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq252_HTML.gif). In terms of polar coordinates, the desired solutions will be obtained if and only if there exists s ˆ ] 0 , 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq228_HTML.gif such that
      θ ( T , s ˆ ) = θ ( s ˆ ) ( 2 j 1 ) π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ19_HTML.gif
      (2.14)

      for some integer j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq239_HTML.gif. Then the proof can be concluded as above, via the intermediate value theorem. We observe that the case of heteroclinic solutions from P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif is analogous, thus it will be omitted. □

      The next theorem deals with the internal connections. For this result, it is useful to recall the rotational gap defined in (2.11).

      Theorem 2.2 (Internal connections)

      Under the above assumptions, the following results hold for solutions which satisfy the energy condition
      x ( t ) 2 + 2 k x ( t ) 2 μ 0 x ( t ) 4 [ 1 , ] , t R : http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equad_HTML.gif
      • there always exist a solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifand a solution homoclinic to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif;

      • there always exist a heteroclinic solution from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifto P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifand a heteroclinic solution from P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifto P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif;

      • if t + t > τ 1 τ τ τ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq253_HTML.gif, there exist at least 2 w ( t + t ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq254_HTML.gifsolutions homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifand 2 w ( t + t ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq254_HTML.gifsolutions homoclinic to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif;

      • if t + t > τ 1 τ τ τ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq255_HTML.gif, there exist at least 2 w ( t + t ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq254_HTML.gifheteroclinic solutions from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gifto P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifand 2 w ( t + t ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq254_HTML.giffrom P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gifto P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif.

      Proof As the first step, we focus our attention on the search of solutions homoclinic to the point P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif. The case of homoclinics to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif is analogous, thus it will be omitted.

      Our goal is to find points on L = L ( P + , P ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq256_HTML.gif which are moved by the Poincaré map Φ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq207_HTML.gif on L = L ( P , P + ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq257_HTML.gif.

      As a preliminary remark, we note that Φ 0 T ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq258_HTML.gif is a simple arc contained in ℰ, connecting its end points Φ 0 T ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq259_HTML.gif and Φ 0 T ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq260_HTML.gif on Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif through the ‘intermediate’ point Φ 0 T ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq261_HTML.gif on Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq262_HTML.gif. Observe also that Φ 0 T ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq259_HTML.gif and Φ 0 T ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq260_HTML.gif are antipodal. In fact, P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif and P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif are antipodal and Φ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq207_HTML.gif is an odd map (indeed, ( x ( t ) , y ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq92_HTML.gif is a solution of (2.7) if and only if ( x ( t ) , y ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq263_HTML.gif is a solution of the same equation). Now, if the trivial situation { Φ 0 T ( P ) , Φ 0 T ( P + ) } = { P , P + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq264_HTML.gif occurs, we have Φ 0 T ( L ) L = { P , P + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq265_HTML.gif. Otherwise, we find that Φ 0 T ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq260_HTML.gif and Φ 0 T ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq259_HTML.gif belong to the two different components of E L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq266_HTML.gif. Since Φ 0 T ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq260_HTML.gif and Φ 0 T ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq259_HTML.gif are the end points of the arc Φ 0 T ( L ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq267_HTML.gif, by an elementary connectivity argument, we conclude that Φ 0 T ( L ) L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq268_HTML.gif. This proves the first assertion of the theorem.

      For the search of heteroclinic solutions from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif, we adopt a similar procedure, choosing as a target the set ℒ itself. The fact that, for every T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq269_HTML.gif, there is always at least one intersection point between Φ 0 T ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq258_HTML.gif and ℒ follows by the same argument developed for homoclinic solutions. Namely, if the trivial situation { Φ 0 T ( P ) , Φ 0 T ( P + ) } = { P , P + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq264_HTML.gif occurs, we have Φ 0 T ( L ) L = { P , P + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq270_HTML.gif. Otherwise, we find that Φ 0 T ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq260_HTML.gif and Φ 0 T ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq259_HTML.gif belong to the two different components of E L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq271_HTML.gif. Therefore, it follows that Φ 0 T ( L ) L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq272_HTML.gif.

      We study now the problem of multiplicity of solutions. As before, we consider at first the case of solutions homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif.

      We introduce in the inner region ℰ of a phase-plane for (2.7) a system of modified polar coordinates ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif with center in the origin. In this system, every point is determined by its angular coordinate θ and its energy E defined in (2.8). Since the energy level lines in ℰ are strictly star-shaped with respect to the origin, we obtain a coordinate system equivalent to the polar one.

      Observe that the set L L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq274_HTML.gif intersects every half-line from the origin exactly in one point. Accordingly, we can parameterize those points using the angular coordinate. With this convention, the initial points in the arc ℒ are parameterized as γ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq226_HTML.gif for s [ π , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq275_HTML.gif (s is the angle) with γ ( π ) = P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq276_HTML.gif, γ ( 2 π ) = P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq277_HTML.gif and γ ( 3 π / 2 ) = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq278_HTML.gif. As a consequence, E ( γ ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq222_HTML.gif is strictly decreasing for s [ π , 3 π / 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq279_HTML.gif and strictly increasing in [ 3 π / 2 , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq280_HTML.gif. The target set L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq281_HTML.gif (the symmetric of ℒ with respect to the x-axis) is parameterized with the angle s [ 0 , π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq282_HTML.gif.

      For the specific case of system (2.6), an analytic expression for ℒ as a graph in the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane, is given as follows:
      E ( θ ) : = E ( ρ ( θ ) cos θ , ρ ( θ ) sin θ ) = ρ 2 ( θ ) sin 2 θ + 2 k ρ 2 ( θ ) cos 2 θ μ 0 ρ 4 ( θ ) cos 4 θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ20_HTML.gif
      (2.15)
      with
      ρ ( θ ) = ( 2 k / μ 1 ) ( sin 2 θ + 4 k cos 2 θ ) 1 / 2 sin θ , θ [ π , 2 π ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equae_HTML.gif

      Hence, the natural parametrization of ℒ in the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane is given by γ ( s ) = ( s , E ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq283_HTML.gif for s = θ [ π , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq284_HTML.gif.

      Using the same modified polar coordinates to represent the solutions of (2.7), we can express the final points Φ 0 T ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq258_HTML.gif in the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane by means of their angular coordinate θ ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq285_HTML.gif and energy E ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq286_HTML.gif. As in the previous proof, we observe that, for every s [ π , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq275_HTML.gif, the map t θ ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq287_HTML.gif is strictly decreasing on [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq6_HTML.gif, while the energy is constant with respect to t. Observe also that in the new coordinates ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif, system (2.7) becomes
      { θ = sin 2 θ 2 k cos 2 θ + 2 μ 0 Λ ( θ , E ) cos 4 θ , E = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ21_HTML.gif
      (2.16)
      with
      Λ ( θ , E ) : = 2 E sin 2 θ + 2 k cos 2 θ + ( ( sin 2 θ + 2 k cos 2 θ ) 2 4 μ 0 E cos 4 θ ) 1 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equaf_HTML.gif
      In the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane, the points of ℒ move with a negative angular speed along the lines of constant energy. Thus, under the action of the flow associated to (2.16), the points of ℒ shift from the right to the left in the strip
      E ( , 2 π ] : = { ( θ , E ) : θ 2 π , 1 E } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equag_HTML.gif
      As we have explained before, we obtain an internal solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif whenever there is a point z L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq288_HTML.gif such that Φ 0 T ( z ) L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq289_HTML.gif, with T = t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq200_HTML.gif. In the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane, this target set is expressed as the union of the graphs
      Ξ : = j = 0 { ( θ , ϒ ( θ ) ) : 2 j π θ π 2 j π } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equah_HTML.gif
      where
      ϒ ( θ ) : = E ( ϱ ( θ ) cos θ , ϱ ( θ ) sin θ ) = ϱ 2 ( θ ) sin 2 θ + 2 k ϱ 2 ( θ ) cos 2 θ μ 0 ϱ 4 ( θ ) cos 4 θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equai_HTML.gif
      with
      ϱ ( θ ) = ( 2 k / μ 1 ) ( sin 2 θ + 4 k cos 2 θ ) 1 / 2 + sin θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equaj_HTML.gif
      (see Figures 10, 11 for a graphical description).
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig10_HTML.jpg
      Figure 10

      The initial set http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq179_HTML.gif and the target set L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq290_HTML.gifrepresented in the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq291_HTML.gif-plane. The present figure is drawn for the parameters k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq292_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq126_HTML.gif. The internal region ℰ corresponds to the strip R × [ 1 , ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq293_HTML.gif. Since we are interested in the evolution of the set ℒ through the Poincaré map, we consider only the half-strip E ( , 2 π ] = ( , 2 π ] × [ 1 , ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq294_HTML.gif. For the flow associated to (2.16), all the points of ℒ move from the right to the left on lines parallel to the θ-axis. The point R (as well as the points on the line E = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq295_HTML.gif) moves faster than the points P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq45_HTML.gif which are on the line E = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq296_HTML.gif. During all the evolution, the distance between the images of P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif and P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif remains constantly equal to π.

      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig11_HTML.jpg
      Figure 11

      The present figure describes the same situation of Figure 7 in the angle-energy coordinates (with the same choice of the coefficients k , μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq297_HTML.gif , μ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq298_HTML.gif and time gap T = t + t = 50 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq299_HTML.gif ).

      In this setting, we obtain an internal solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif if and only if there exists s ˆ [ π , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq300_HTML.gif such that Φ ( γ ( s ˆ ) ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq301_HTML.gif, where we have denoted by Φ the Poincaré map for (2.16) in the time interval [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq302_HTML.gif. Of course Φ is exactly Φ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq207_HTML.gif (which was the Poincaré map associated to (2.7)) in the new ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-coordinates. Using the parameterized curve γ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq226_HTML.gif for the initial points, we can express Φ as
      Φ ( γ ( s ) ) = ( θ ( T , s ) , E ( s ) ) for  s [ π , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equak_HTML.gif

      (recall that γ ( s ) = ( s , E ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq283_HTML.gif for s = θ [ π , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq303_HTML.gif).

      The points P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif and P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif are antipodal, lie on the same energy line Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif and move with the same angular speed. In the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane, P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif and P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif are expressed by P = ( π , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq304_HTML.gif, P + = ( 2 π , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq305_HTML.gif. Therefore,
      θ ( t , π ) θ ( t , 2 π ) = π , t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ22_HTML.gif
      (2.17)
      Moreover, for every integer j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq306_HTML.gif, we have that
      θ ( t , π ) π j π if and only if t j τ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equal_HTML.gif

      (this follows from the fact that the orbit Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif has a period τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq102_HTML.gif and is symmetric with respect to the x-axis).

      In the same plane, the point R is indicated by R = ( 3 π / 2 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq307_HTML.gif. Using the fact that the orbit Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif has a period τ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq308_HTML.gif and is symmetric with respect to the y-axis, we find that, for every integer i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq309_HTML.gif,
      θ ( t , 3 π / 2 ) π 2 2 i π if and only if t τ 1 2 + i τ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equam_HTML.gif
      Suppose now that
      2 T τ 1 2 T τ > 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ23_HTML.gif
      (2.18)

      Then, the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq204_HTML.gif contains at least one odd integer.

      Let m o ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq310_HTML.gif be the largest odd integer contained in ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq201_HTML.gif. From T > m o ( T ) τ 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq311_HTML.gif, we have that
      θ ( T , 3 π / 2 ) < π 2 ( m o ( T ) 1 ) π = 3 π 2 m o ( T ) π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ24_HTML.gif
      (2.19)
      Suppose also that j is a positive integer with 2 T / τ < j m o ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq312_HTML.gif. In this case, from T < j τ / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq313_HTML.gif, we have that
      θ ( T , π ) > π j π and θ ( T , 2 π ) > 2 π j π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ25_HTML.gif
      (2.20)
      Let
      L left : = { ( θ , E ( θ ) ) : π θ 3 π / 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equan_HTML.gif
      and
      L right : = { ( θ , E ( θ ) ) : 3 π / 2 θ 2 π } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equao_HTML.gif
      with L = L left L right http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq314_HTML.gif be the left and the right parts of ℒ in the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane. Similarly, we define
      L left : = { ( θ , ϒ ( θ ) ) : 0 θ π / 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equap_HTML.gif
      and
      L right : = { ( θ , ϒ ( θ ) ) : π / 2 θ π } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equaq_HTML.gif

      By definition, for any nonnegative integer i, the set L left ( 2 i π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq315_HTML.gif is a simple arc connecting the points ( 2 i π , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq316_HTML.gif and ( 2 i π + ( π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq317_HTML.gif in the rectangle [ 2 i π , 2 i π + ( π / 2 ) ] × [ 1 , ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq318_HTML.gif and, similarly, L right ( 2 i π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq319_HTML.gif is a simple arc connecting the points ( 2 i π + ( π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq317_HTML.gif and ( 2 i π + π , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq320_HTML.gif in the rectangle [ 2 i π + ( π / 2 ) , 2 i π + π ] × [ 1 , ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq321_HTML.gif. On the other hand, the set Φ ( L left ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq322_HTML.gif is a simple arc connecting the points ( θ ( T , π ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq323_HTML.gif and ( θ ( T , 3 π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq324_HTML.gif in E ( , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq325_HTML.gif and, similarly, Φ ( L right ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq326_HTML.gif is a simple arc connecting the points ( θ ( T , 3 π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq324_HTML.gif and ( θ ( T , 2 π ) , ) = ( π + θ ( T , π ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq327_HTML.gif in the same strip.

      Suppose now that j = m o ( T ) = 2 k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq328_HTML.gif (in this case, j is odd). In such a situation, from (2.19) and (2.20) we have that
      θ ( T , 3 π / 2 ) < π 2 2 k π , θ ( T , π ) > 2 k π , θ ( T , π ) > π 2 k π , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equar_HTML.gif

      for 2 k = j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq329_HTML.gif an even integer. Hence the points ( θ ( T , 3 π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq324_HTML.gif and ( θ ( T , π ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq323_HTML.gif are separated by the arc L left ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq330_HTML.gif, while the points ( θ ( T , 3 π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq324_HTML.gif and ( θ ( T , 2 π ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq331_HTML.gif are separated by L left ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq330_HTML.gif and L right ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq332_HTML.gif. As a consequence, Φ ( L ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq333_HTML.gif contains at least three points, precisely the nonempty intersections of Φ ( L left ) ( L left ( 2 k π , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq334_HTML.gif, Φ ( L right ) ( L left ( 2 k π , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq335_HTML.gif and Φ ( L right ) ( L right ( 2 k π , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq336_HTML.gif.

      Suppose that j = m o ( T ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq337_HTML.gif (in this case, j is even). In such a situation, from (2.19) and (2.20) we have that
      θ ( T , 3 π / 2 ) < π 2 2 k π , θ ( T , π ) > π 2 k π , θ ( T , π ) > 2 π 2 k π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equas_HTML.gif

      for 2 k = m o ( T ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq338_HTML.gif an even integer. Hence the points ( θ ( T , 3 π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq324_HTML.gif and ( θ ( T , π ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq323_HTML.gif are separated by the arcs L left ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq330_HTML.gif and L right ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq332_HTML.gif. The points ( θ ( T , 3 π / 2 ) , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq324_HTML.gif and ( θ ( T , 2 π ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq331_HTML.gif are separated by L left ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq330_HTML.gif, L right ( 2 k π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq332_HTML.gif and L left ( ( 2 k 1 ) π , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq339_HTML.gif. As a consequence, Φ ( L ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq333_HTML.gif contains at least five points: two points coming from the intersections of Φ ( L left ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq322_HTML.gif with the target set Ξ and three from the intersections of Φ ( L right ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq326_HTML.gif.

      Proceeding by induction, with the same argument, the following result is obtained.

      Claim 1 Φ ( L ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq340_HTML.gif has at least 2 n # + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq341_HTML.gif solutions, where n # http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq342_HTML.gif is the number of integers less than or equal to m o ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq310_HTML.gif which are contained in the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq201_HTML.gif.

      A completely symmetric argument leads to the same multiplicity result for solutions which are homoclinic to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif.

      At last, we look for a multiplicity result for heteroclinic solutions from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif. We follow step by step the argument described in the part of the proof devoted to the search of multiple internal homoclinic solutions and, therefore, we transform our equation in the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-coordinates. As we have explained before, we obtain an internal heteroclinic (from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif) whenever there is a point z L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq288_HTML.gif such that Φ 0 T ( z ) L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq343_HTML.gif, with T = t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq344_HTML.gif. In the ( θ , E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq273_HTML.gif-plane, this target set is expressed as the union of the graphs
      Ξ : = j = 0 { ( θ , E ( θ ) ) : π 2 j π θ 2 π 2 j π } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equat_HTML.gif

      where E ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq345_HTML.gif is the function defined in (2.15).

      As before, we assume the validity of condition (2.18). Then the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq346_HTML.gif contains at least one even integer.

      Let m e ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq347_HTML.gif be the largest even integer contained in ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq201_HTML.gif. From T > m e ( T ) τ 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq348_HTML.gif, we have that
      θ ( T , 3 π / 2 ) < 3 π 2 m e ( T ) π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equau_HTML.gif

      For any positive integer j with 2 T / τ < j m e ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq349_HTML.gif, we have that (2.20) holds too. At this point, we have simply to repeat (with obvious changes due to the fact that the target set is Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq350_HTML.gif which is a shift of Ξ by ( 2 p i , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq351_HTML.gif) the argument presented above for the case of homoclinic connections and obtain the following conclusion.

      Claim 2 Φ ( L ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq352_HTML.gif has at least 2 n # + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq353_HTML.gif solutions, where n # http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq354_HTML.gif is the number of integers less than or equal to m e ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq347_HTML.gif which are contained in the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq201_HTML.gif.

      A completely symmetric argument leads to a multiplicity result for heteroclinic solutions from P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif.

      As final step, we have just to show how the condition on the rotational gap allows us to achieve the conclusion from Claim 1 and Claim 2 that we have obtained along the proof. Now, it is sufficient to observe that t + t = T > τ 1 τ τ τ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq355_HTML.gif implies (2.18) and, therefore, the open interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq204_HTML.gif contains at least two integers. More precisely, by the definition of w ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq356_HTML.gif, the interval ] 2 T / τ , 2 T / τ 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq204_HTML.gif contains at least w ( T ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq205_HTML.gif integers and, therefore,
      n # , n # w ( T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equav_HTML.gif

      This concludes the proof of the theorem.  □

      3 Remarks and related results

      We end the paper with a list of remarks about possible variants and extensions of the main results obtained for equation
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equaw_HTML.gif
      1. 1.

        A problem which naturally arises from the analysis that we have performed concerns what happens if, for a stepwise weight function satisfying (2.5), we suppose that 0 < μ 1 < μ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq357_HTML.gif. Repeating the preliminary phase-plane analysis of Section 2.2, one can easily check that for any gap T = t + t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq358_HTML.gif, there always exist a solution homoclinic to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif and another homoclinic to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq48_HTML.gif, as well as a heteroclinic from P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq33_HTML.gif to P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif and another one from P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq32_HTML.gif to P + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq51_HTML.gif. However, in general, if 0 < μ 1 < μ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq359_HTML.gif, one cannot obtain multiplicity results like those achieved in Section 2 without some extra assumptions on h ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq360_HTML.gif. This is the reason for which, in the study of equation (2.1), we have considered only the case 0 < μ 0 < μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq361_HTML.gif. For different examples on related equations in which a weight coefficient can be above or below its limits at infinity, see, for instance, [46].

         
      On the other hand, if we assume that the weight function changes its sign, with μ 0 < 0 < μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq362_HTML.gif, some interesting multiplicity results could be produced. In fact the phase portrait of (1.1) in the time interval [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif shows a global center, hence if t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq27_HTML.gif is large, a lot of connections between the unstable and stable manifolds of the points P ± http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq45_HTML.gif can be obtained. For the sake of conciseness, we omit the study of this latter situation, which is beyond the goal of the present paper.
      1. 2.
        We notice that the same argument of the proofs applies to an equation of the form
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equax_HTML.gif
         
      for σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq3_HTML.gif, h ( x ) = x f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq363_HTML.gif, and with f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq364_HTML.gif satisfying condition (∗), if we assume that a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq25_HTML.gif is a stepwise weight function, playing the same role of g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif in (2.1). In such a case, the saddle points become P ± = ( ± ξ 1 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq365_HTML.gif with ξ 1 : = f 1 ( σ / μ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq366_HTML.gif, where f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq367_HTML.gif is the inverse of f restricted to [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq6_HTML.gif. In this way, we can apply our results to nonlinearities like those considered in [2, 27]. More precisely, Theorem 2.1 holds without any further assumption, while for Theorem 2.2 we need to require a gap between the periods of the orbits Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq74_HTML.gif and Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq77_HTML.gif. In order to obtain this gap, we can apply (for instance) some results ensuring the monotonicity of the time-map (like [[28], Theorem A]).
      1. 3.
        Due to the special form of the weight coefficient, it is standard to verify (via a simple rescaling procedure) that (1.1) is equivalent to an equation of the form
        x + b ( t ) x a h ( x ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ26_HTML.gif
        (3.1)
         
      where a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq368_HTML.gif and b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq369_HTML.gif is a stepwise coefficient. In this manner, one can deal with some nonlinear Schrödinger-type equations related to the case of potential wells or potential walls [29, 30].
      1. 4.

        The approach used in the proofs, based on the properties of the Poincaré map, guarantees that our results are stable with respect to small perturbations. More precisely, fixed a suitable length T = t + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq370_HTML.gif for the time interval, Theorems 2.1 and 2.2 provide a lower bound for the number of solutions. We can state that the same lower bound persists for a small perturbation of the coefficient in the L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq371_HTML.gif-norm on [ t , t + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq26_HTML.gif. Therefore, the assumption that the weight a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq25_HTML.gif in (1.1) or b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq369_HTML.gif in (3.1) are stepwise functions can be slightly relaxed, so we can ‘smooth’ them.

         
      2. 5.
        With reference to equations (1.1) or (3.1) with stepwise coefficients, we observe that our approach can be adapted to boundary value problems on a compact interval [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq302_HTML.gif like, e.g., the Dirichlet (two-point) or the Neumann problem. In these cases, we have to find solutions connecting given lines which depart from the origin. For the sake of conciseness, we cannot describe the most general situation, but we just outline a possible application for the Neumann problem
        { x + 2 k x 2 g ( t ) x 3 = 0 , x ( 0 ) = 0 , x ( T ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equ27_HTML.gif
        (3.2)
         
      with g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq2_HTML.gif a stepwise function such that g ( t ) = μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq372_HTML.gif for t [ 0 , τ [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq373_HTML.gif and g ( t ) = μ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq374_HTML.gif for t [ τ , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq375_HTML.gif, for 0 < μ 0 < μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq376_HTML.gif. Let us denote by Φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq377_HTML.gif the Poincaré map associated to system (2.6) on the time interval [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq378_HTML.gif, and by Φ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq379_HTML.gif the Poincaré map associated to system (2.7) on the time interval [ 0 , T τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq380_HTML.gif. We observe that there exists a maximal compact interval [ ξ 1 , ξ 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq381_HTML.gif with ξ 1 < ξ 1 < ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq382_HTML.gif such that E ( Φ 1 ( z ) ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq383_HTML.gif for all the points z = ( x , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq384_HTML.gif with x [ ξ 1 , ξ 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq385_HTML.gif. In this manner Φ 1 ( ± ξ 1 , 0 ) Γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq386_HTML.gif. The curve
      ϒ : = { Φ 1 ( x , 0 ) : ξ 1 x ξ 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Equay_HTML.gif
      represents the set of all the points in the region W = { ( x , y ) : ξ 0 x ξ 0 , E ( x , y ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq387_HTML.gif, which are images (by Φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq377_HTML.gif) of initial points of the x-axis (therefore x ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq388_HTML.gif). Figure 12 illustrates the curve ϒ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq389_HTML.gif for a short time interval ( τ = 2.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq390_HTML.gif). For a larger τ, the line ϒ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq389_HTML.gif becomes a double spiral with a certain number of turns around the origin, while the part of ϒ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq389_HTML.gif contained in the region ℱ gets very close to O ( P + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq96_HTML.gif and O ( P ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq391_HTML.gif. Then we can repeat the same argument developed in the previous sections by looking for the intersections of Φ 2 ( ϒ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq392_HTML.gif with the x-axis.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_Fig12_HTML.jpg
      Figure 12

      In the present figure the line ϒ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq393_HTML.gifis the transformation of the segment [ ξ 1 , ξ 1 ] × { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq394_HTML.gifunder the Poincaré map Φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq395_HTML.giffor τ = 2.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq396_HTML.gif. The parameters used are k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq85_HTML.gif, μ 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq125_HTML.gif, μ 0 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-167/MediaObjects/13661_2012_Article_422_IEq126_HTML.gif. For graphical reasons, a slightly different x- and y-scaling has been used.

      1. 6.

        As a final remark, we mention the fact that combining our technique with Ważewski’s method [31], following the approach developed by Conley in [18], one can deal with some more general classes of weight functions. For example, one could tackle with these techniques the cases of asymptotically constant or asymptotically periodic coefficients. These extensions, however, need a more delicate analysis and they are beyond the goals of the present paper.

         

      We hope that the abundance of multiplicity results found in the present work (in the special case of stepwise coefficients) may suggest possible directions for extending Theorem 2.1 and Theorem 2.2 to more general weight functions. This will be our goal for a future investigation of the problem.

      Declarations

      Acknowledgements

      The authors are deeply indebted with the referee for the careful checking of the manuscript and for his/her remarks, including a correction to an erroneous argument in the previous version of the proof. This research was partially supported by the project PRIN-2009 Equazioni Differenziali Ordinarie e Applicazioni.

      Authors’ Affiliations

      (1)
      Department of Mathematics and Computer Science, University of Udine

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