Melnikov theory for weakly coupled nonlinear RLC circuits

Flaviano Battelli; Michal Fečkan

doi:10.1186/1687-2770-2014-101

Research
Open Access

Melnikov theory for weakly coupled nonlinear RLC circuits

Flaviano Battelli¹Email author and
Michal Fečkan^{2, 3}

Boundary Value Problems20142014:101

https://doi.org/10.1186/1687-2770-2014-101

Received: 18 December 2013
Accepted: 1 April 2014
Published: 7 May 2014

Abstract

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some coupled implicit differential equations. In particular we show the persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds. Application is given to coupled nonlinear RLC system.

MSC: Primary 34A09; secondary 34C23; 37G99.

Keywords

implicit ode
perturbation
Melnikov method
RLC circuits

1 Introduction

In [1], motivated by [2, 3], the equation modeling nonlinear RLC circuits

{(u + f (u))}^{″} + ε γ {(u + f (u))}^{'} + u + ε h (t + α, u, ε) = 0

(1)

has been studied. It is assumed that

f (u)

and

h (t, u, ε)

are smooth functions with

f (u)

at least quadratic at the origin and satisfying suitable assumptions. Setting

v = {(u + f (u))}^{'}

the equation reads

\begin{aligned} (1 + f^{'} (u)) u^{'} = v, \\ v^{'} = - u - ε [h (t + α, u, ε) + γ v] . \end{aligned}

(2)

It is assumed that, for some

u_{0} \in R

, we have

f^{'} (u_{0}) + 1 = 0

and

u_{0} f^{″} (u_{0}) < 0

. So for

ε = 0

(2) has the Hamiltonian

H (u, v) = v^{2} + 2 \int_{u_{0}}^{u} σ (1 + f^{'} (σ)) d σ

passing through

(u_{0}, 0)

. Clearly

\nabla H (u_{0}, 0) = (\begin{array}{c} 0 \\ 0 \end{array})

and the Hessian of ℋ at

(u_{0}, 0)

is

H_{H} (u_{0}, 0) = (\begin{array}{c} 2 u_{0} f^{″} (u_{0}) & 0 \\ 0 & 2 \end{array}),

so that the condition

u_{0} f^{″} (u_{0}) < 0

means that

(u_{0}, 0)

is a saddle for ℋ. Multiplying the second equation by

1 + f^{'} (u)

we get the system

(1 + f^{'} (u)) (\begin{array}{c} u^{'} \\ v^{'} \end{array}) = (\begin{array}{c} v \\ - (1 + f^{'} (u)) {u + ε [h (t + α, u, ε) + γ v]} \end{array}) .

(3)

Note that (2) falls in the class of implicit differential equations (IODE) like

A (x) x^{'} = f (x) + ε h (t, x, ε, κ), (ε, κ) \in R \times R^{m},

(4)

with $A (u, v) = (\begin{array}{c} 1 + f^{'} (u) & 0 \\ 0 & 1 \end{array})$ . Obviously, $det A (u, v) = 1 + f^{'} (u)$ vanishes on the line $(u_{0}, v)$ and the condition $f^{″} (u_{0}) \neq 0$ implies that the line $u = u_{0}$ consists of noncritical 0-singularities for (3) (see [[4], p.163]). Let $N L$ denote the kernel of the linear map L and $R L$ its range. Then $R A (u_{0}, 0)$ is the subspace having zero first component and then the right hand side of (3) belongs to $R A (u_{0}, 0)$ if and only if $v = 0$ . So all the singularities $(u_{0}, v)$ with $v \neq 0$ are impasse points while $(u_{0}, 0)$ is a so called I-point (see [[4], pp.163-166]). Quasilinear implicit differential equations, such as (4), find applications in a large number of physical sciences and have been studied by several authors [4–12]. On the other hand, there are many other works on implicit differential equations [13–18] dealing with more general implicit differential systems by using analytical and topological methods.

Passing from (2) to (3), in the general case, it corresponds to multiplying (4) by the adjugate matrix

A^{a} (x)

:

ω (x) x^{'} = A^{a} (x) [f (x) + ε h (t, x, ε, κ)],

where $ω (x) = det A (x)$ . Here we note that A and x may have different dimensions in this paper depending on the nature of the equation but the concrete dimension is clear from that equation, so we do not use different notations for A and x. Basic assumptions in [1] are $ω (x_{0}) = 0$ , $ω^{'} (x_{0}) \neq 0$ and $A^{a} (x_{0}) f (x_{0}) = 0$ , $A^{a} (x_{0}) h (t, x_{0}, ε, κ) = 0$ for some $x_{0}$ (that is, $x_{0}$ is an I-point for (4)) and the existence of a solution $x (t)$ in a bounded interval J tending to $x_{0}$ as t tends to the endpoints of J.

It is well known [4, 8] that

ω (x_{0}) = 0

and

ω^{'} (x_{0}) \neq 0

imply

dim N A (x_{0}) = 1, R A^{a} (x_{0}) = N A (x_{0}), and N A^{a} (x_{0}) = R A (x_{0}),

(5)

and then $A^{a} (x_{0}) f (x_{0}) = 0$ is equivalent to the fact that $f (x_{0}) \in R A (x_{0})$ .

Let $F (x) : = A^{a} (x) f (x)$ . It has been proved in [19] that (5) implies that $rank F^{'} (x_{0})$ is at most 2. So, if $x \in R^{n}$ , with $n > 2$ then $x = x_{0}$ cannot be hyperbolic for the map $x \mapsto F^{'} (x_{0}) x$ .

In this paper we study coupled IODEs such as

\begin{aligned} A_{0} (x_{1}) x_{1}^{'} = f (x_{1}) + ε g_{1} (t, x_{1}, x_{2}, ε, κ), \\ A_{0} (x_{2}) x_{2}^{'} = f (x_{2}) + ε g_{2} (t, x_{1}, x_{2}, ε, κ), \end{aligned}

(6)

with

x_{1}, x_{2} \in R^{2}

,

det A_{0} (x_{0}) = 0 \neq {(det A_{0})}^{'} (x_{0})

,

f (x_{0}), g_{j} (t, x_{0}, x_{0}, ε, κ) \in R A_{0} (x_{0})

and other assumptions that will be specified below. Let us remark that (6) is a special kind of the general equation (4) with, among other things,

A (x) = (\begin{array}{c} A_{0} (x_{1}) & 0 \\ 0 & A_{0} (x_{2}) \end{array}), x = (x_{1}, x_{2})

hence

det A (x) = det A_{0} (x_{1}) det A_{0} (x_{2})

satisfies

det A (x_{0}, x_{0}) = 0

,

{(det A)}^{'} (x_{0}, x_{0}) = 0

and

{(det A)}^{″} (x_{0}, x_{0}) \neq 0

. Thus

(x_{0}, x_{0})

is not a I-point. Multiplying the first equation by

A_{0}^{a} (x_{1})

and the second by

A_{0}^{a} (x_{2})

we obtain the system

\begin{aligned} ω (x_{1}) x_{1}^{'} = F (x_{1}) + ε G_{1} (x_{1}, x_{2}, t, ε, κ), \\ ω (x_{2}) x_{2}^{'} = F (x_{2}) + ε G_{2} (x_{1}, x_{2}, t, ε, κ) . \end{aligned}

(7)

We assume that

ω (x) : = det A_{0} (x)

,

F (x)

and

G_{j} (x_{1}, x_{2}, t, ε, κ)

satisfy the following assumptions:

(C1)

$F \in C^{2} (R^{2}, R^{2})$ , $ω \in C^{2} (R^{2}, R)$ and the unperturbed equation
$ω (x) x^{'} = F (x)$

(8)

possesses a noncritical singularity at $x_{0}$ , i.e. $ω (x_{0}) = 0$ and $ω^{'} (x_{0}) \neq 0$ .
(C2)

$F (x_{0}) = 0$ and the spectrum $σ (F^{'} (x_{0})) = {μ_{\pm}}$ with $μ_{-} < 0 < μ_{+}$ , and
$x^{'} = F (x)$

has a solution $γ (s)$ homoclinic to $x_{0}$ , that is, ${lim}_{s \to \pm \infty} γ (s) = x_{0}$ , and $ω (γ (s)) \neq 0$ for any $s \in R$ . Without loss of generality, we may, and will, assume $ω (γ (s)) > 0$ for any $s \in R$ . Moreover, $G_{i} \in C^{2} (R^{6 + m}, R^{2})$ , $i = 1, 2$ are 1-periodic in t with $G_{i} (x_{0}, x_{0}, t, ε, κ) = 0$ for any $t \in R$ , $κ \in R^{m}$ and ε sufficiently small.
(C3)

Let $γ_{\pm}$ be the eigenvectors of $F^{'} (x_{0})$ with the eigenvalues $μ_{\mp}$ , resp. Then $〈 \nabla ω (x_{0}), γ_{\pm} 〉 > 0$ (or else $ω^{'} (x_{0}) γ_{\pm} > 0$ ).

From (C2) we see that

Γ (s) : = (\begin{array}{c} γ (s) \\ γ (s) \end{array})

is a bounded solution of the equation

\begin{aligned} x_{1}^{'} = F (x_{1}), \\ x_{2}^{'} = F (x_{2}) \end{aligned}

(9)

and that $x_{0}$ persists as a singularity of (7). So this paper is a continuation of [1, 19], but here we study more degenerate IODE.

The objective of this paper is to give conditions, besides (C1)-(C3), assuring that for $| ε | ≪ 1$ , the coupled equations (7) has a solution in a neighborhood of the orbit ${Γ (s) ∣ s \in R}$ and reaching $(x_{0}, x_{0})$ is a finite time. Our approach mimics that in [1] and uses Melnikov methods to derive the needed conditions. Let us briefly describe the content of this paper. In Section 2 we make few remarks concerning assumptions (C1)-(C3). Then, in Section 3, we change time to reduce equation (7) to a smooth perturbation of (9) whose unperturbed part has the solution $Γ (s)$ . Next, in Section 4 we derive the Melnikov condition. Finally Section 5 is devoted to the application of our result to coupled equations of the form (1) for RLC circuits, while some computations are postponed to the appendix.

We emphasize the fact that Melnikov technique is useful to predict the existence of transverse homoclinic orbits in mechanical systems [20, 21] together with the associated chaotic behavior of solutions. However, the result in this paper is somewhat different in that we apply the method to show existence of orbits connecting a singularity in finite time.

2 Comments on the assumptions

By following [1, 19] we note that since

γ (s) \to x_{0}

as

| s | \to \infty

then

γ^{'} (s)

is a bounded solution of the linear equation

x^{'} = F (γ (s)) x

. Hence

| γ^{'} (s) | \leq k e^{- μ | s |}

for some

μ > 0

. We get then, for

s \geq 0

,

| γ (s) - x_{0} | \leq \int_{s}^{\infty} | γ^{'} (s) | d s \leq μ^{- 1} k e^{- μ s} .

So

\underset{s \to \infty}{lim sup} \frac{log | γ (s) - x_{0} |}{s} \leq - μ < 0 .

From [[22], Theorem 4.3, p.335 and Theorem 4.5, p.338] it follows that

\underset{s \to \infty}{lim sup} \frac{log | γ (s) - x_{0} |}{s} = μ_{-} < 0,

(10)

and there exist a constant

δ > 0

and a solution

γ_{+} e^{μ_{-} s}

of

x^{'} = F^{'} (x_{0}) x

such that

| γ (s) - x_{0} - γ_{+} e^{μ_{-} s} | = O (e^{(μ_{-} - δ) s}), as s \to \infty .

Note that

γ_{+} \neq 0

since otherwise

γ (s) - x_{0} = O (e^{(μ_{-} - δ) s})

, contradicting (10). Hence

γ_{+}

is an eigenvector of the eigenvalue

μ_{-}

of

F^{'} (x_{0})

. We have then

| \frac{γ (s) - x_{0}}{e^{μ_{-} s}} - γ_{+} | \leq c_{1} e^{- δ s}

for a suitable constant

c_{1} \geq 0

. As a consequence,

lim_{s \to \infty} \frac{γ (s) - x_{0}}{e^{μ_{-} s}} = γ_{+} .

| γ_{+} | - c_{1} e^{- δ s} \leq \frac{| γ (s) - x_{0} |}{e^{μ_{-} s}} \leq | γ_{+} | + c_{1} e^{- δ s} .

Taking logarithms, dividing by s and letting

s \to \infty

we get

lim_{s \to \infty} \frac{log | γ (s) - x_{0} |}{s} = μ_{-},

that is, in (10)

{lim sup}_{s \to \infty}

can be replaced with

{lim}_{s \to \infty}

. Similarly, changing s with −s:

lim_{s \to - \infty} \frac{log | γ (s) - x_{0} |}{s} = μ_{+} .

Next, set

φ (s) : = \frac{1}{e^{- μ_{-} s} + e^{- μ_{+} s}} .

(11)

Since

\frac{φ (s)}{e^{μ_{-} s}} \to 1

as

s \to \infty

and

\frac{φ (s)}{e^{μ_{+} s}} \to 1

as

s \to - \infty

we have then

lim_{s \to \pm \infty} \frac{γ (s) - x_{0}}{φ (s)} = γ_{\pm} \neq 0

(12)

and

lim_{s \to \pm \infty} \frac{ω (γ (s))}{φ (s)} = lim_{s \to \pm \infty} \frac{ω^{'} (x_{0}) (γ (s) - x_{0}) + o (γ (s) - x_{0})}{e^{μ_{\mp} s}} = 〈 \nabla ω (x_{0}), γ_{\pm} 〉 .

From (C2), we know $ω (γ (s)) > 0$ for any $s \in R$ , so $〈 \nabla ω (x_{0}), γ_{\pm} 〉 \geq 0$ . Hence condition (C3) means that $γ (s)$ tends transversally to the singular manifold $ω^{- 1} (0)$ at $x_{0}$ .

As in [19] it is easily seen that

lim_{s \to \pm \infty} \frac{γ^{'} (s)}{ω (γ (s))} = \frac{F^{'} (x_{0}) γ_{\pm}}{ω^{'} (x_{0}) γ_{\pm}} = \frac{μ_{\mp} γ_{\pm}}{ω^{'} (x_{0}) γ_{\pm}} \neq 0

(13)

and that

\frac{γ^{'} (s)}{ω (γ (s))}

solves the equation

x^{'} = [F^{'} (γ (s)) - \frac{F (γ (s))}{ω (γ (s))} ω^{'} (γ (s))] x = F^{'} (γ (s)) x - \frac{ω^{'} (γ (s)) x}{ω (γ (s))} F (γ (s)) .

(14)

So

\frac{γ^{'} (s)}{ω (γ (s))}

is a bounded solution of (14). Next, setting as in [19]

θ (s) : = \int_{0}^{s} ω (γ (τ)) d τ

(15)

and

x_{h} (t) = γ (θ^{- 1} (t))

, it is easily seen that

x_{h} (t)

satisfies

ω (x) x^{'} = F (x)

whose linearization along

x_{h} (t)

is

F^{'} (x_{h} (t)) z = x_{h}^{'} (t) ω^{'} (x_{h} (t)) z + ω (x_{h} (t)) z^{'} = F (x_{h} (t)) \frac{ω^{'} (x_{h} (t)) z}{ω (x_{h} (t))} + ω (x_{h} (t)) z^{'}

i.e.

ω (x_{h} (t)) z^{'} = F^{'} (x_{h} (t)) z - F (x_{h} (t)) \frac{ω^{'} (x_{h} (t)) z}{ω (x_{h} (t))} .

(16)

Note, then, that (14) is derived from (16) with the change $x (s) = z (θ (s))$ . This fact should clarify why we need to consider the linear system (14) instead of $x^{'} = F^{'} (γ (s)) x$ . However, see [19] for a remark concerning the space of bounded solutions of (14) and that of the equation $x^{'} = F^{'} (γ (s)) x$ .

We now prove that $\frac{γ^{'} (s)}{ω (γ (s))}$ is the unique solution of equation (14) which is bounded on ℝ. This is a kind of nondegeneracy of $γ (s)$ .

Lemma 2.1 Assume (C2) and (C3) hold. Then, up to a multiplicative constant, $\frac{γ^{'} (s)}{ω (γ (s))}$ is the unique solution of (14) which is bounded on ℝ.

Proof From [[19], Lemma 3.1] it follows that the linear map:

x \mapsto [F^{'} (x_{0}) - \frac{μ_{-} γ_{+}}{ω^{'} (x_{0}) γ_{+}} ω^{'} (x_{0}) - μ_{-} I] x

has the simple eigenvalues

μ_{+} - μ_{-}

and

- μ_{-}

. Let

μ : = \frac{μ_{+}}{2}

, then the linear map

x \mapsto [F^{'} (x_{0}) - \frac{μ_{-} γ_{+}}{ω^{'} (x_{0}) γ_{+}} ω^{'} (x_{0}) - μ I] x

has the eigenvalues ±μ; moreover, since

c_{1} \leq \frac{| γ^{'} (s) |}{ω (γ (s))} \leq c_{2}

for two positive constants

0 < c_{1} < c_{2}

, it follows that

γ_{0} (s) : = \frac{γ^{'} (s)}{ω (γ (s))} e^{- μ s}

is a solution of

x^{'} = [F^{'} (γ (s)) - \frac{F (γ (s))}{ω (γ (s))} ω^{'} (γ (s)) - μ I] x

(17)

satisfying

\frac{c_{1}}{c_{2}} | γ_{0} (s_{1}) | \leq | γ_{0} (s_{2}) | e^{μ (s_{2} - s_{1})} \leq \frac{c_{2}}{c_{1}} | γ_{0} (s_{1}) |

for all

0 \leq s_{1} \leq s_{2}

. Then (17) satisfies the assumptions of [[19], Theorem 5.3] and hence its conclusion with

rank P_{+} = 1

, that is, the fundamental matrix

X_{+} (s)

of (17) satisfies

\begin{matrix} ∥ X_{+} (s_{2}) P_{+} X_{+}^{- 1} (s_{1}) ∥ \leq k e^{- μ (s_{2} - s_{1})}, 0 \leq s_{1} \leq s_{2}, \\ ∥ X_{+} (s_{2}) (I - P_{+}) X_{+}^{- 1} (s_{1}) ∥ \leq k e^{\tilde{μ} (s_{2} - s_{1})}, 0 \leq s_{2} \leq s_{1}, \end{matrix}

where $0 \leq \tilde{μ} < μ$ . However, it is well known (see [23–25]) that $R P_{+}$ is the space of initial conditions for the bounded solutions on $[0, \infty [$ of (17) that, then, tend to zero as $s \to \infty$ at the exponential rate $e^{- μ s}$ . As a consequence a solution $u (s)$ of (17) is bounded on $[0, \infty [$ if and only if $u (s) e^{μ s}$ is a bounded solution of (14). Then we conclude that the space of solutions of (14) that are bounded on $[0, \infty [$ is one dimensional.

Incidentally, since the fundamental matrix of (14) is

X (s) = X_{+} (s) e^{μ s}

, we note that it satisfies

\begin{matrix} ∥ X (s_{2}) P_{+} X^{- 1} (s_{1}) ∥ \leq k, 0 \leq s_{1} \leq s_{2}, \\ ∥ X (s_{2}) (I - P_{+}) X^{- 1} (s_{1}) ∥ \leq k e^{(μ + \tilde{μ}) (s_{2} - s_{1})}, 0 \leq s_{2} \leq s_{1} . \end{matrix}

Using a similar argument in

R_{-} =] - \infty, 0]

with

μ = \frac{μ_{-}}{2} < 0

, and [[19], Theorem 5.4] instead of [[19], Theorem 5.3] with

μ^{*} = - μ

we see that (14) has at most a one dimensional space of solutions bounded in ℝ. More precisely,

\tilde{μ}

with

μ < \tilde{μ} < 0

, and a projection

P_{-}

on

R^{2}

exists such that

\begin{matrix} ∥ X (s_{2}) (I - P_{-}) X^{- 1} (s_{1}) ∥ \leq k, s_{2} \leq s_{1} \leq 0, \\ ∥ X (s_{2}) P_{-} X^{- 1} (s_{1}) ∥ \leq k e^{(μ + \tilde{μ}) (s_{2} - s_{1})}, s_{1} \leq s_{2} \leq 0, \end{matrix}

and $dim N P_{-} = 1$ . Since $\frac{γ^{'} (s)}{ω (γ (s))}$ is a solution of (14) bounded on ℝ we deduce that $R P_{+} = N P_{-} = span {\frac{γ^{'} (0)}{ω (γ (0))}}$ and the result follows. □

We conclude this section with a remark about condition (c) in [[19], Theorem 5.3]. Consider a system in

R^{n}

such as

x^{'} = [D + A (s)] x .

(18)

Then the following result holds.

Theorem 2.2 Suppose the following hold:

(i)

D has two simple eigenvalues $μ_{*} < μ^{*}$ and all the other eigenvalues of D have either real part less than $μ_{*}$ or greater than $μ^{*}$ ;
(ii)

$\int_{0}^{\infty} ∥ A (s) ∥ d s < \infty$ ;
(iii)

$A (s) \to 0$ as $s \to \infty$ .

Then there are as many solutions

x (t)

of (18) satisfying

k_{1} | x (s) | \leq | x (t) | e^{- μ_{*} (t - s)} \leq k_{2} | x (s) |, for any 0 \leq s \leq t,

(19)

as the dimension of the space of the generalized eigenvectors of the matrix D with real parts less than or equal to

μ_{*}

; here

k_{1}, k_{2} > 0

are two suitable positive constants. Similarly there are as many solutions of (18) such that

{\tilde{k}}_{1} | x (s) | \leq | x (t) | e^{- μ^{*} (t - s)} \leq {\tilde{k}}_{2} | x (s) |, for any 0 \leq s \leq t,

(20)

for suitable constants ${\tilde{k}}_{1}, {\tilde{k}}_{2} > 0$ , as the dimension of the space of the generalized eigenvectors of the matrix D with real parts greater than or equal to $μ^{*}$ .

Proof We prove the first statement concerning (19). By a similar argument (20) is handled. Changing variables we may assume that

D = (\begin{array}{c} μ_{*} & 0 & 0 \\ 0 & D_{-} & 0 \\ 0 & 0 & D_{+} \end{array})

and the eigenvalues of

D_{-}

have real parts less than

μ_{*}

and those of

D_{+}

have real parts greater than or equal to

μ^{*}

. So the system reads

\begin{aligned} x_{1}^{'} = μ_{*} x_{1} + a_{11} (t) x_{1} + A_{12} (t) x_{2} + A_{13} (t) x_{3}, \\ x_{2}^{'} = D_{-} x_{2} + A_{21} (t) x_{1} + A_{22} (t) x_{2} + A_{23} (t) x_{3}, \\ x_{3}^{'} = D_{+} x_{3} + A_{31} (t) x_{1} + A_{32} (t) x_{2} + A_{33} (t) x_{3}, \end{aligned}

(21)

where

a_{11} (t) \in R

and

A_{i j} (t)

are matrices (or vectors) of suitable orders. Setting

y_{i} (t) = e^{- μ_{*} t} x_{i} (t)

we get

\begin{aligned} y_{1}^{'} = a_{11} (t) y_{1} + A_{12} (t) y_{2} + A_{13} (t) y_{3}, \\ y_{2}^{'} = (D_{-} - μ_{*} I) y_{2} + A_{21} (t) y_{1} + A_{22} (t) y_{2} + A_{23} (t) y_{3}, \\ y_{3}^{'} = (D_{+} - μ_{*} I) y_{3} + A_{31} (t) y_{1} + A_{32} (t) y_{2} + A_{33} (t) y_{3} . \end{aligned}

(22)

Now we observe that

y (t)

is a solution of (22) bounded at +∞ if and only if

x (t)

is a solution of (21) which is bounded on ℝ when multiplied by

e^{- μ_{*} t}

. Moreover, since

| a_{11} (t) |

,

| A_{12} (t) |

, and

| A_{13} (t) |

belong to

L^{1} (R)

, the limit

{lim}_{t \to + \infty} y_{1} (t)

exists for any solution

y (t)

of (22) bounded on

R_{+}

. So, let us fix

t_{0} > 0

and take

t \geq t_{0}

. If

y (t)

is a solution of (22) bounded at +∞ it must be, by the variation of constants formula,

\begin{aligned} y_{1} (t) = y_{1}^{\infty} - \int_{t}^{\infty} (a_{11} (s) y_{1} (s) + A_{12} (s) y_{2} (s) + A_{13} (s) y_{3} (s)) d s, \\ y_{2} (t) = e^{(D_{-} - μ_{*} I) (t - t_{0})} y_{2}^{0} \\ + \int_{t_{0}}^{t} e^{(D_{-} - μ_{*} I) (t - s)} (A_{21} (s) y_{1} (s) + A_{22} (s) y_{2} (s) + A_{23} (s) y_{3} (s)) d s, \\ y_{3} (t) = - \int_{t}^{\infty} e^{(D_{+} - μ_{*} I) (t - s)} (A_{31} (s) y_{1} (s) + A_{32} (s) y_{2} (s) + A_{33} (s) y_{3} (s)) d s, \end{aligned}

(23)

where

y_{2}^{0} = y_{2} (t_{0})

and

y_{1}^{\infty} = {lim}_{t \to + \infty} y_{1} (t)

. Note that since

σ (D_{-} - μ_{*} I) \subset {λ \in C ∣ ℜ λ < 0}

and

σ (D_{+} - μ_{*} I) \subset {λ \in C ∣ ℜ λ > 0}

and

a_{11} (t)

,

A_{i j} (t)

are bounded, we can interpret (22) as a fixed point theorem in the Banach space of bounded function on

[t_{0}, + \infty [

:

B : = {y (t) \in C^{0} ([t_{0}, \infty [) ∣ sup | y (t) | < \infty}

with the obvious norm. Since

a_{11} (t), A_{i j} (t) \in L^{1} (R_{+})

we see that the map (23) is a contraction on B, provided

t_{0}

is sufficiently large, and then, for any given

(y_{1}^{\infty}, y_{2}^{0})

, it has a unique solution

y (t, y_{1}^{\infty}, y_{2}^{0}) \in B

. Note that a priori

y (t, y_{1}^{\infty}, y_{2}^{0})

is defined only on

[t_{0}, + \infty [

but of course we can extend it to

[0, + \infty [

going backward with time. We now prove that positive constants

0 < c_{1} < c_{2}

exist such that

c_{1} \leq | y (t, y_{0}) | \leq c_{2}

fox any

t \geq 0

. Let

t_{0} < t_{1} < t

. We have

\begin{aligned} y_{2} (t_{1}) = & e^{(D_{-} - μ_{*} I) (t_{1} - t_{0})} y_{2}^{0} \\ + \int_{t_{0}}^{t_{1}} e^{(D_{-} - μ_{*} I) (t_{1} - s)} (A_{21} (s) y_{1} (s) + A_{22} (s) y_{2} (s) + A_{23} (s) y_{3} (s)) d s \end{aligned}

and then

\begin{aligned} y_{2} (t) = & e^{(D_{-} - μ_{*} I) (t - t_{0})} y_{2}^{0} + \int_{t_{0}}^{t_{1}} e^{(D_{-} - μ_{*} I) (t - s)} (A_{21} (s) y_{1} (s) + A_{22} (s) y_{2} (s) + A_{23} (s) y_{3} (s)) d s \\ + \int_{t_{1}}^{t} e^{(D_{-} - μ_{*} I) (t - s)} (A_{21} (s) y_{1} (s) + A_{22} (s) y_{2} (s) + A_{23} (s) y_{3} (s)) d s \\ = & e^{(D_{-} - μ_{*} I) (t - t_{1})} y_{2} (t_{1}) + \int_{t_{1}}^{t} e^{(D_{-} - μ_{*} I) (t - s)} (A_{21} (s) y_{1} (s) + A_{22} (s) y_{2} (s) + A_{23} (s) y_{3} (s)) d s . \end{aligned}

So, for any

δ > 0

let

t_{1}

be such that

{sup}_{t \geq t_{1}} | A_{i j} (t) | \leq δ

and set

{sup}_{t \geq t_{1}} | y_{j} (t) | = {\bar{y}}_{j}

. We have

\begin{aligned} {\bar{y}}_{2} & = sup_{t \geq t_{1}} | y_{2} (t) | \leq k e^{α (t - t_{1})} {\bar{y}}_{2} + \int_{t_{1}}^{t} k e^{α (t - s)} δ ({\bar{y}}_{1} + {\bar{y}}_{2} + {\bar{y}}_{3}) d s \\ \leq k ({\bar{y}}_{2} + \frac{δ}{α} ({\bar{y}}_{1} + {\bar{y}}_{2} + {\bar{y}}_{3})) e^{α (t - t_{1})} + \frac{k δ}{| α |} ({\bar{y}}_{1} + {\bar{y}}_{2} + {\bar{y}}_{3}) \end{aligned}

with

max {ℜ μ ∣ μ \in σ (D_{-} - μ_{*} I)} < α < 0

. Taking the limit as

t \to + \infty

we get

{\bar{y}}_{2} \leq \frac{δ}{| α |} ({\bar{y}}_{1} + {\bar{y}}_{2} + {\bar{y}}_{3}) .

Since

δ \to 0

as

t_{1} \to + \infty

, from the above it follows that

{lim}_{t \to \infty} | y_{2} (t) | = 0

. Similarly we get

| y_{3} | \leq k δ β^{- 1} ({\bar{y}}_{1} + {\bar{y}}_{2} + {\bar{y}}_{3}),

where

0 < β < min {ℜ μ ∣ μ \in σ (D_{+} - μ_{*} I)}

and then

{lim}_{t \to \infty} | y_{3} (t) | = 0

. As a consequence we obtain

{lim}_{t \to \infty} | y (t) | - | y_{1} (t) | = 0

and then

lim_{t \to \infty} | y (t) | = | y_{1}^{\infty} | .

So, provided we take

y_{1}^{\infty} \neq 0

we see that eventually (i.e. for

t \geq \bar{t}

, for some

\bar{t} > 0

)

\frac{| y_{1}^{\infty} |}{2} \leq | y (t) | \leq \frac{3}{2} | y_{1}^{\infty} |

and the existence of

c_{1}, c_{2} > 0

such that

c_{1} \leq | y (t) | \leq c_{2}

for all

t \geq 0

follows from the fact that

| y (t) |

cannot vanish in any bounded interval. Finally since

| x (t) | = | y (t) | e^{μ_{*} t}

we get, for

0 \leq s \leq t

,

\frac{| x (t) |}{| x (s) |} = \frac{| y (t) |}{| y (s) |} e^{μ_{*} (t - s)} \Rightarrow \frac{c_{1}}{c_{2}} e^{μ_{*} (t - s)} \leq \frac{| x (t) |}{| x (s) |} \leq \frac{c_{2}}{c_{1}} e^{μ_{*} (t - s)}

i.e.

\frac{c_{1}}{c_{2}} | x (s) | \leq | x (t) | e^{- μ_{*} (t - s)} \leq \frac{c_{2}}{c_{1}} | x (s) | .

The proof is complete. □

Remark 2.3 (i) It follows from the proof of Theorem 2.2 that inequalities of (19) also hold replacing (i) with the weaker assumption that

μ_{*}

is a simple eigenvalue of D and all the others either have real parts less than

μ_{*}

or

\geq μ^{*}

(i.e. we do not need that

μ^{*}

is simple). Similarly inequalities of (20) hold if

μ^{*}

is a simple eigenvalue of D and all the others either have real parts greater than

μ^{*}

or

\leq μ_{*}

(i.e. we do not need that

μ_{*}

is simple).

(ii)

Note that a result related to Theorem 2.2 has been proved in [26].

3 Solutions asymptotic to the fixed point

It follows from (11)-(12) that

γ (s) - x_{0} = O (e^{μ_{\mp} s})

as

s \to \pm \infty

then, since

γ^{'} (s) = F (γ (s)) = O (γ (s) - x_{0})

we obtain

γ^{'} (s) = O (e^{μ_{\mp} s})

. Furthermore, from (13) we also get:

ω (γ (s)) = O (γ^{'} (s)) = O (e^{μ_{\mp} s}) .

As a consequence

T_{\pm} : = \int_{0}^{\pm \infty} ω (γ (τ)) d τ < \infty .

Since

ω (γ (s)) > 0

it follows that

θ : R \to] T_{-}, T_{+} [

is a strictly increasing diffeomorphism (see (15) for the definition of

θ (s)

). Then

x_{h} (t) : = γ (θ^{- 1} (t))

satisfies (8) on the interval

] T_{-}, T_{+} [

and

lim_{t \to T_{\pm}} x_{h} (t) = x_{0} .

Moreover (see (13))

lim_{t \to t_{\pm}} x_{h}^{'} (t) = lim_{t \to t_{\pm}} \frac{F (x_{h} (t))}{ω (x_{h} (t))} = lim_{s \to \pm \infty} \frac{F (γ (s))}{ω (γ (s))} = μ_{\mp} \frac{γ_{\pm}}{ω^{'} (x_{0}) γ_{\pm}} \neq 0 .

Hence

x_{0}

is not an I-point of (8). In this paper we want to look for solutions of the coupled equation (7) that belong to a neighborhood of

{(x_{h} (t), x_{h} (t)) ∣ T_{-} < t < T_{+}}

, they are defined in the interval

] T_{-} + α, T_{+} + α [

, for some

α = α (ε)

, and tend to

(x_{0}, x_{0})

at the same rate as

(x_{h} (t), x_{h} (t))

. To this end we first perform a change of the time variable as follows. Set

t = α + θ (s) \in] T_{-} + α, T_{+} + α [

and plug

z_{j} (s) = x_{j} (α + θ (s))

in (7). We get

ω (z_{j}) z_{j}^{'} = ω (γ (s)) (F (z_{j}) + ε G_{j} (z_{1}, z_{2}, α + θ (s), ε, κ)), j = 1, 2 .

(24)

Since we are looking for solutions of (7) tending to

(x_{0}, x_{0})

at the same rate as

γ (s)

, in (24) we make the change of variables

z_{j} (s) = γ (s) + φ (s) y_{j} (s) = x_{0} + φ (s) (η (s) + y_{j} (s)), j = 1, 2,

(25)

where

η (s)

is the bounded function

\frac{γ (s) - x_{0}}{φ (s)}

. Since

\begin{aligned} ω (x_{0} + φ (s) (η (s) + y)) \\ \geq 〈 \nabla ω (x_{0}), φ (s) (η (s) + y) 〉 - K_{1} {| φ (s) (η (s) + y) |}^{2} \\ = φ (s) [〈 \nabla ω (x_{0}), η (s) + y 〉 - K_{1} φ (s) {| η (s) + y |}^{2}] \end{aligned}

(26)

for a suitable constant

K_{1} > 0

and any

s \in R

,

| y | \leq 1

we get, using (C3), (26):

ω (x_{0} + φ (s) (η (s) + y)) \geq \frac{1}{2} φ (s) 〈 \nabla ω (x_{0}), γ_{\pm} 〉 > 0

(27)

for

| s | > 0

large and

| y | < δ

sufficiently small. Then (27) and

ω (γ (t)) > 0

imply the existence of

M > 0

and

δ > 0

so that

ω (x_{0} + φ (s) (η (s) + y)) \geq M φ (s)

for any

s \in R

and

| y | \leq δ

. Now plugging (25) into (24) we derive the equations

\begin{aligned} y_{j}^{'} = & \frac{ω (γ (s))}{φ (s) ω (γ (s) + φ (s) y_{j})} F (γ (s) + φ (s) y_{j}) - \frac{F (γ (s))}{φ (s)} - \frac{φ^{'} (s)}{φ (s)} y_{j} \\ + ε \frac{ω (γ (s))}{φ (s) ω (γ (s) + φ (s) y_{j})} G_{j} (γ (s) + φ (s) y_{1}, γ (s) + φ (s) y_{2}, θ (s) + α, ε, κ), \\ j = 1, 2 . \end{aligned}

(28)

From (11) it follows that

\frac{φ^{'} (s)}{φ (s)} = \frac{μ_{-} e^{- μ_{-} (s)} + μ_{+} e^{- μ_{+} (s)}}{e^{- μ_{-} (s)} + e^{- μ_{+} (s)}} \to μ_{\mp} as s \to \pm \infty .

Next we note that from

G_{j} (x_{0}, x_{0}, t, ε, κ) = 0

it follows that the quantities

\begin{aligned} \frac{G_{j} (γ (s) + φ (s) y_{1}, γ (s) + φ (s) y_{2}, α + θ (s), ε, κ)}{φ (s)} \\ = \frac{G_{j} (x_{0} + φ (s) (η (s) + y_{1}), x_{0} + φ (s) (η (s) + y_{2}), α + θ (s), ε, κ)}{φ (s)}, j = 1, 2 \end{aligned}

are bounded uniformly in $s \in R$ and $κ \in R^{m}$ , $y_{1}$ , $y_{2}$ , ε bounded.

The linearization of (28) at

y = 0

,

ε = 0

is

y_{j}^{'} = [F^{'} (γ (s)) - \frac{F (γ (s)) ω^{'} (γ (s))}{ω (γ (s))} - \frac{φ^{'} (s)}{φ (s)} I] y_{j}, j = 1, 2 .

(29)

Taking the limit as

s \to + \infty

we get the systems

y_{j}^{'} = [F^{'} (x_{0}) - \frac{μ_{-} γ_{+}}{ω^{'} (x_{0}) γ_{+}} ω^{'} (x_{0}) - μ_{-} I] y_{j}, j = 1, 2 .

(30)

Similarly taking the limit as

s \to - \infty

we get the systems

y_{j}^{'} = [F^{'} (x_{0}) - \frac{μ_{+} γ_{-}}{ω^{'} (x_{0}) γ_{-}} ω^{'} (x_{0}) - μ_{+} I] y_{j}, j = 1, 2 .

(31)

From the proof of Lemma 2.1 (see also [[1], Lemma 3.1]) we know that (30) has the positive simple eigenvalues

μ_{+} - μ_{-}

and

- μ_{-}

, and (31) has the negative simple eigenvalues

μ_{-} - μ_{+}

and

- μ_{+}

. From the roughness of exponential dichotomies it follows that both equations in (29) have an exponential dichotomy on both

R_{+}

and

R_{-}

with projections, resp.

P_{+} = 0

and

P_{-} = I

. Hence (see also [19]) all solutions of the system

y^{'} = - [F^{'} {(γ (s))}^{*} - \frac{ω^{'} {(γ (s))}^{*} F {(γ (s))}^{*}}{ω (γ (s))} - \frac{φ^{'} (s)}{φ (s)} I] y,

(32)

adjoint to (29), are bounded as $| s | \to \infty$ . We let $ψ_{1} (s)$ and $ψ_{2} (s)$ be any two linearly independent solutions of (32).

4 Melnikov function and the original equation

In this section we will give a condition for solving (28) for

y_{1} (t), y_{2} (t)

near the solution

y_{1} (t) = y_{2} (t) = 0

of the same equation with

ε = 0

. Writing

F (y) : = y^{'} - \frac{ω (γ (s))}{φ (s) ω (γ (s) + φ (s) y)} F (γ (s) + φ (s) y) + \frac{F (γ (s))}{φ (s)} + \frac{φ^{'} (s)}{φ (s)} y

(33)

and

\begin{aligned} H_{j} (y_{1}, y_{2}, η, ε) : = & - \frac{ω (γ (s))}{φ (s) ω (γ (s) + φ (s) y_{j})} \\ \times G_{j} (γ (s) + φ (s) y_{1}, γ (s) + φ (s) y_{2}, θ (s) + α, ε, κ), j = 1, 2, \end{aligned}

(34)

we look for solutions

y_{1} (t), y_{2} (t) : R \to R^{2}

of

\begin{array}{r} F (y_{1}) + ε H_{1} (y_{1}, y_{2}, η, ε) = 0, \\ F (y_{2}) + ε H_{2} (y_{1}, y_{2}, η, ε) = 0 \end{array}

(35)

in the Banach space of

C^{1}

-functions on ℝ, bounded together with their derivatives and with small norms. We observe that

F (0) = 0

and equation

F^{'} (0) y = 0

reads

y^{'} = [F^{'} (γ (s)) - \frac{F (γ (s)) ω^{'} (γ (s))}{ω (γ (s))} - \frac{φ^{'} (s)}{φ (s)}] y .

(36)

In Section 3 (see also [1, 19]) we have seen that (36) has an exponential dichotomy on both $R_{+}$ and $R_{+}$ with projections $P_{+} = I - P_{-} = 0$ . So the only bounded solution $y (t)$ of $F^{'} (0) y = 0$ is $y (t) = 0$ . In other words ${NF}^{'} (0) = {0}$ . So we are lead to prove the following.

Theorem 4.1 Let Y, X be Banach spaces,

ε \in R

be a small parameter and

η \in R^{2 d}

. Let

F : Y \to X

,

H_{1, 2} : Y \times Y \times R^{2 d + 1} \to X

,

(y_{1}, y_{2}, η, ε) \mapsto H_{1, 2} (y_{1}, y_{2}, η, ε)

be

C^{2}

-functions such that

(a)

$F (0) = 0$ ;
(b)

${NF}^{'} (0) = {0}$ ;
(c)

there exist $ψ_{1}, \dots, ψ_{d} \in X^{*}$ such that ${RF}^{'} (0) = {ψ_{1}, \dots, ψ_{d}}^{\circ}$ .

Set

M : R^{2 d} \to R^{2 d}

by

M (η) : = (\begin{array}{c} ψ_{1} H_{1} (0, 0, 0, η, 0) \\ ⋮ \\ ψ_{d} H_{1} (0, 0, 0, η, 0) \\ ψ_{1} H_{2} (0, 0, 0, η, 0) \\ ⋮ \\ ψ_{d} H_{2} (0, 0, 0, η, 0) \end{array})

(37)

and suppose there exists

\bar{η} \in R^{2 d}

such that

M (\bar{η}) = 0

and the derivative

M^{'} (\bar{η})

is invertible. Then there exist

r > 0

and unique

C^{1}

-function

η = η (ε)

, defined in a neighborhood of

ε = 0 \in R

such that

lim_{ε \to 0} η (ε) = \bar{η}

(38)

and for

η = η (ε)

,

ε \neq 0

, (35) has a unique solution

(y_{1} (ε), y_{2} (ε)) \in Y \times Y

satisfying

∥ (y_{1} (ε), y_{2} (ε)) ∥ \leq r .

Moreover,

y_{j} (ε) = ε {\tilde{y}}_{j} (ε)

for

C^{0}

-functions

{\tilde{y}}_{j} (ε) \in Y

and we have

F^{'} (0) {\tilde{y}}_{j} (0) + H_{j} (0, 0, 0, \bar{η}, 0) = 0, j = 1, 2 .

(39)

Proof We look for solutions

(y_{1}, y_{2}, η)

of (35) that are close to

(y_{1}, y_{2}, η) = (0, 0, \bar{η})

. Let

P : X \to X

be the projection such that

R P = {RF}^{'} (0)

. Note

codim {RF}^{'} (0) = d

. From the implicit function theorem, we solve the projected equations

\begin{matrix} P F (y_{1}) + ε P H_{1} (y_{1}, y_{2}, η, ε) = 0, \\ P F (y_{2}) + ε P H_{2} (y_{1}, y_{2}, η, ε) = 0 \end{matrix}

for unique

y_{1, 2} = Y_{1, 2} (η, ε) \in Y

such that

Y_{1, 2} (η, 0) = 0,

provided

| ε | \leq ε_{0}

is sufficiently small and η in a fixed closed ball

Ξ \subset R^{2 d}

with

\bar{η} \in \overset{\circ}{Ξ}

. Note that

Y_{1, 2}

are

C^{2}

-smooth. Setting

Q = I - P

, we need to solve the bifurcation equations:

Q F (Y_{j} (η, ε)) + ε Q H_{j} (Y_{1} (η, ε), Y_{2} (η, ε), η, ε) = 0, j = 1, 2 .

(40)

Observe that

Q x = 0 \Leftrightarrow x \in R P = {RF}^{'} (0) \Leftrightarrow ψ_{i} x = 0 for all i = 1, \dots, d .

(41)

Then

Q F^{'} (0) = 0

and so

Q F (Y_{j} (η, ε)) = O_{j} (ε^{2}), j = 1, 2,

uniformly with respect to η. We conclude that (40) can be written as

ε Q H_{j} (0, 0, η, 0) = - Q R_{j} (η, ε), j = 1, 2,

(42)

where

R_{j} (η, ε) : = F (Y_{j} (η, ε)) + ε [H_{j} (Y_{1} (η, ε), Y_{2} (η, ε), η, ε) - H_{j} (0, 0, η, 0)] .

Note that

R_{j} (η, ε)

are

C^{2}

-functions of

(η, ε)

and that

ε^{- 1} R_{j} (η, ε) = O (ε)

uniformly with respect to η, so

{\tilde{R}}_{j} (η, ε) : = {\begin{cases} - ε^{- 1} R_{j} (η, ε), & if ε \neq 0, \\ 0, & if ε = 0 \end{cases}

is

C^{1}

in

(η, ε)

. By (41), system (42) is equivalent to

M (η) = {(\begin{array}{c} ψ_{i} {\tilde{R}}_{1} (η, ε) \\ ψ_{i} {\tilde{R}}_{2} (η, ε) \end{array})}_{i = 1, \dots, d} = O (ε) .

(43)

Because of the assumptions we can apply the implicit function theorem to (43) to obtain a

C^{1}

-function

η (ε)

defined in a neighborhood of

ε = 0

satisfying (43) and such that (38) holds. Setting

y_{j} (ε) : = Y_{j} (η (ε), ε), j = 1, 2

we see that

y_{1} (ε)

,

y_{2} (ε)

are bounded

C^{1}

-solutions of (35) with

η = η (ε)

such that

y_{1} (0) = 0

,

y_{2} (0) = 0

. Then we can write

y_{1} (ε) = ε {\tilde{y}}_{1} (ε)

,

y_{2} (ε) = ε {\tilde{y}}_{2} (ε)

) for continuous

{\tilde{y}}_{1} (ε), {\tilde{y}}_{2} (ε) \in Y

where

F (ε {\tilde{y}}_{j} (ε)) + ε H_{j} (ε {\tilde{y}}_{1} (ε), ε {\tilde{y}}_{2} (ε), η (ε), ε) = 0, j = 1, 2 .

Clearly (39) follows differentiating the above equality at $ε = 0$ . The proof is complete. □

Remark 4.2 Note that, because of

M (\bar{η}) = 0

, (39) is equivalent to

P F^{'} (0) {\tilde{y}}_{j} (0) + H_{j} (0, 0, 0, \bar{η}, 0) = 0, j = 1, 2,

which has the unique solution

{\tilde{y}}_{j} (0) = - {[P F^{'} (0)]}^{- 1} H_{j} (0, 0, 0, \bar{η}, 0), j = 1, 2 .

(44)

Now we apply Theorem 4.1 to (28) with

F (y)

,

H_{1} (y_{1}, y_{2}, η, ε)

,

H_{2} (y_{1}, y_{2}, η, ε)

as in (33), (34) and

Y = C_{b}^{1} (R, R^{2}), X = C_{b}^{0} (R, R^{2}), η = (α, κ) \in R \times R^{m},

where $C_{b}^{k} (R, R^{2})$ is the Banach space of $C^{k}$ -functions bounded together with their derivatives with the usual sup-norm.

We already observed that

F (0) = 0

and

{NF}^{'} (0) = 0

. Moreover,

{RF}^{'} (0) = {x = (x_{1}, x_{2}) \in X | \int_{- \infty}^{\infty} ψ_{i}^{*} (s) x_{i} (s) d s = 0, i = 1, 2},

where

ψ_{i} (s)

have been defined in the previous Section 3. So

d = 2

and

m = 3

. We recall, from [19], that

ψ_{j} (s) = φ (s) v_{j} (θ (s))

where

v_{j} (t)

are solutions of the adjoint equation of (16):

ω (x_{h} (t)) v^{'} = \frac{ω^{'} {(x_{h} (t))}^{*}}{ω (x_{h} (t))} F {(x_{h} (t))}^{*} v - F^{'} {(x_{h} (t))}^{*} v, t \in] T_{-}, T_{+} [,

(45)

and

v_{j} (0) = R P_{-}^{*} \cap N P_{+}^{*}

. Hence (37) reads

M (α, κ) = (\begin{array}{c} \int_{- \infty}^{\infty} \frac{ψ_{1}^{*} (s)}{φ (s)} G_{1} (γ (s), γ (s), α + θ (s), 0, κ) d s \\ \int_{- \infty}^{\infty} \frac{ψ_{2}^{*} (s)}{φ (s)} G_{1} (γ (s), γ (s), α + θ (s), 0, κ) d s \\ \int_{- \infty}^{\infty} \frac{ψ_{1}^{*} (s)}{φ (s)} G_{2} (γ (s), γ (s), α + θ (s), 0, κ) d s \\ \int_{- \infty}^{\infty} \frac{ψ_{2}^{*} (s)}{φ (s)} G_{2} (γ (s), γ (s), α + θ (s), 0, κ) d s \end{array})

or passing to time

t = θ (s)

:

M (α, κ) = (\begin{array}{c} \int_{T_{-}}^{T_{+}} v_{1}^{*} (t) \frac{G_{1} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \\ \int_{T_{-}}^{T_{+}} v_{2}^{*} (t) \frac{G_{1} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \\ \int_{T_{-}}^{T_{+}} v_{1}^{*} (t) \frac{G_{2} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \\ \int_{T_{-}}^{T_{+}} v_{2}^{*} (t) \frac{G_{2} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \end{array}) .

(46)

A direct application of Theorem 4.1 gives the following.

Theorem 4.3 Let

m = 3

and

M (α, κ)

be given as in (46) where

v_{1} (t)

,

v_{2} (t)

are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose that

\bar{α}

and

\bar{κ}

exist so that

M (\bar{α}, \bar{κ}) = 0 and \frac{\partial M}{\partial (α, κ)} (\bar{α}, \bar{κ}) \in G L (4, R) .

(47)

Then there exist

\bar{ε} > 0

,

ρ > 0

, unique

C^{1}

-functions

α (ε)

and

κ (ε)

with

α (0) = \bar{α}

and

κ (0) = \bar{κ}

, defined for

| ε | < \bar{ε}

, and a unique solution

(z_{1} (s, ε), z_{2} (s, ε))

of (24) with

α = α (ε)

,

κ = κ (ε)

,

0 < | ε | < \bar{ε}

, such that

sup_{s \in R} | z_{j} (s, ε) - γ (s) | φ {(s)}^{- 1} < ρ, j = 1, 2 .

(48)

Moreover,

sup_{s \in R} | z_{j} (s, ε) - γ (s) | φ {(s)}^{- 1} = O (ε), j = 1, 2 .

Remark 4.4 (i) Equation (48) implies

z_{j} (s, ε) = γ (s) + ε {\tilde{y}}_{j} (s, ε) φ (s), j = 1, 2,

for

C^{0}

-functions

{\tilde{y}}_{j} (s, ε)

with

{sup}_{| ε | \leq \bar{ε}} {sup}_{s \in R} | {\tilde{y}}_{j} (s, ε) | < \infty

. Then we have

z_{j} (s, ε) = γ (s) + ε {\tilde{y}}_{j} (s, ε) φ (s) = γ (s) + ε {\tilde{y}}_{j} (s, 0) φ (s) + ε w_{j} (s, ε) φ (s)

with

w_{j} (s, ε) = {\tilde{y}}_{j} (s, ε) - {\tilde{y}}_{j} (s, 0)

, so

{lim}_{ε \to 0} {sup}_{s \in R} | w_{j} (s, ε) | = 0

. Hence

lim_{ε \to 0} sup_{s \in R} | z_{j} (s, ε) - γ (s) - ε {\tilde{y}}_{j} (s, 0) φ (s) | φ {(s)}^{- 1} ε^{- 1} = 0, j = 1, 2,

(49)

which gives a first order approximation of

z_{j} (s, ε)

. Next,

{\tilde{y}}_{j} (s, 0)

can be computed using (44) adapted to this case. Hence

{\tilde{y}}_{j} (s, 0)

are bounded solutions of

y_{j}^{'} = [F^{'} (γ (s)) - \frac{F (γ (s)) ω^{'} (γ (s))}{ω (γ (s))} - \frac{φ^{'} (s)}{φ (s)}] y_{j} + \frac{1}{φ (s)} G_{j} (γ (s), γ (s), θ (s) + \bar{α}, \bar{κ}) .

Since (36) has exponential dichotomies on both

R_{+}

(with projection

P_{+} = 0

) and

R_{-}

(with projection

P_{-} = I

) it follows that

{\tilde{y}}_{j} (s, 0) = {\begin{cases} - \int_{s}^{\infty} X (s) X^{- 1} (z) \frac{1}{φ (z)} G_{j} (γ (z), γ (z), θ (z) + \bar{α}, \bar{κ}) d z & for s \geq 0, \\ \int_{- \infty}^{s} X (s) X^{- 1} (z) \frac{1}{φ (z)} G_{j} (γ (z), γ (z), θ (z) + \bar{α}, \bar{κ}) d z & for s \leq 0 . \end{cases}

(50)

X (s)

is the fundamental solution of (36). Note formulas (50) are well defined at

s = 0

, i.e.,

{\tilde{y}}_{j} (0^{-}, 0) = {\tilde{y}}_{j} (0^{+}, 0)

, due to the first assumption of (47). Next, passing to time

t = θ (s)

and taking

z_{j} (t) : = φ (θ^{- 1} (t)) {\tilde{y}}_{j} (θ^{- 1} (t), 0)

, we get

\begin{aligned} z_{j} (t) & = \int_{- \infty}^{θ^{- 1} (t)} φ (θ^{- 1} (t)) X (θ^{- 1} (t)) X^{- 1} (z) φ {(z)}^{- 1} G_{j} (γ (z), γ (z), θ (z) + \bar{α}, \bar{κ}) d z \\ = \int_{T_{-}}^{t} φ (θ^{- 1} (t)) X (θ^{- 1} (t)) X^{- 1} (θ^{- 1} (u)) φ {(θ^{- 1} (u))}^{- 1} \frac{G_{j} (x_{h} (u), x_{h} (u), u + \bar{α}, \bar{κ})}{ω (x_{h} (u))} d u \end{aligned}

for

T_{-} < t \leq 0

and

z_{j} (t) = - \int_{t}^{T_{+}} φ (θ^{- 1} (t)) X (θ^{- 1} (t)) X^{- 1} (θ^{- 1} (u)) φ {(θ^{- 1} (u))}^{- 1} \frac{G_{j} (x_{h} (u), x_{h} (u), u + \bar{α}, \bar{κ})}{ω (x_{h} (u))} d u

for

0 \leq t < T_{+}

. Note that

z_{j} (t)

solves

ω (x_{h} (t)) z^{'} = F^{'} (x_{h} (t)) z - F (x_{h} (t)) \frac{ω^{'} (x_{h} (t)) z}{ω (x_{h} (t))} + G_{j} (x_{h} (t), x_{h} (t), t + \bar{α}, \bar{κ})

with ${sup}_{t \in] T_{-}, T_{+} [} | z_{j} (t) | φ {(θ^{- 1} (t))}^{- 1} < \infty$ , and $φ (θ^{- 1} (t)) X (θ^{- 1} (t))$ is a fundamental solution of (16).

(ii) Using (49), the functions

x_{j} (t, ε) : = z_{j} (θ^{- 1} (t - α (ε)), ε)

are bounded solutions of (7) in the interval

] T_{-} + α (ε), T_{+} + α (ε) [

such that

lim_{ε \to 0} sup_{t \in] T_{-}, T_{+} [} | x_{j} (t + α (ε), ε) - x_{h} (t) - ε z_{j} (t) | φ {(θ^{- 1} (t))}^{- 1} ε^{- 1} = 0 .

Summarizing, we obtain the following corollary of Theorem 4.3.

Corollary 4.5 Let

m = 3

and

M (α, κ)

be given as in (46) where

v_{1} (t)

,

v_{2} (t)

are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose that

\bar{α}

and

\bar{κ}

exist so that

M (\bar{α}, \bar{κ}) = 0 and \frac{\partial M}{\partial (α, κ)} (\bar{α}, \bar{κ}) \in G L (4, R) .

Then there exist

\bar{ε} > 0

,

ρ > 0

, unique

C^{1}

-functions

α (ε)

and

κ (ε)

with

α (0) = \bar{α}

and

κ (0) = \bar{κ}

, defined for

| ε | < \bar{ε}

, and a unique solution

(x_{1} (t, ε), x_{2} (t, ε))

of (7) with

α = α (ε)

,

κ = κ (ε)

,

0 < | ε | < \bar{ε}

, such that

sup_{T_{-} < t < T_{+}} | x_{j} (t + α (ε), ε) - x_{h} (t) | φ {(θ^{- 1} (t))}^{- 1} < ρ, j = 1, 2 .

Moreover,

sup_{T_{-} < t < T_{+}} | x_{j} (t + α (ε), ε) - x_{h} (t) | φ {(θ^{- 1} (t))}^{- 1} = O (ε), j = 1, 2 .

5 Applications to RLC circuits

In this section we study the coupled equations

\begin{aligned} {(u_{1} + u_{1}^{2})}^{″} + ε γ_{1} {(u_{1} + u_{1}^{2})}^{'} + u_{1} - ε λ {(u_{2} + u_{2}^{2})}^{″} + ε sin t = 0, \\ {(u_{2} + u_{2}^{2})}^{″} + ε γ_{2} {(u_{2} + u_{2}^{2})}^{'} + u_{2} - ε λ {(u_{1} + u_{1}^{2})}^{″} + ε χ sin (t + ϖ) = 0, \end{aligned}

(51)

which is motivated by [2, 3]. Note that (51) is obtained by coupling two equations modeling nonlinear RLC circuits as in (1). Here

γ_{1}

,

γ_{2}

, λ, χ and ϖ are positive parameters. Setting

w_{j} = {(u_{j} + u_{j}^{2})}^{'}

,

j = 1, 2

, (51) reads

{\begin{cases} (2 u_{1} + 1) u_{1}^{'} = w_{1}, \\ w_{1}^{'} + ε γ_{1} w_{1} + u_{1} - ε λ w_{2}^{'} + ε sin t = 0, \\ (2 u_{2} + 1) u_{2}^{'} = w_{2}, \\ w_{2}^{'} + ε γ_{2} w_{2} + u_{2} - ε λ w_{1}^{'} + ε χ sin (t + ϖ) = 0 . \end{cases}

(52)

By solving the second and fourth equations of (52) for

w_{1}^{'}

and

w_{2}^{'}

, we get:

{\begin{cases} (2 u_{1} + 1) u_{1}^{'} = w_{1}, \\ w_{1}^{'} = - u_{1} + ε \frac{sin t + λ u_{2} + γ_{1} w_{1} + ε λ (χ sin (t + ϖ) + λ u_{1} + γ_{2} w_{2})}{ε^{2} λ^{2} - 1}, \\ (2 u_{2} + 1) u_{2}^{'} = w_{2}, \\ w_{2}^{'} = - u_{2} + ε \frac{χ sin (t + ϖ) + λ u_{1} + γ_{2} w_{2} + ε λ (sin t + λ u_{2} + γ_{1} w_{1})}{ε^{2} λ^{2} - 1}, \end{cases}

(53)

provided

| ε λ | \neq 1

. Since

ω (u, w) = ω (u) = 2 u + 1

, to write the system (52) in the form (7) with parameter

κ = (γ_{1}, γ_{2}, λ)

and

(χ, ϖ)

fixed, we have to multiply the second and fourth equation by

2 u_{1, 2} + 1

, respectively, and we obtain the system

{\begin{cases} (2 u_{1} + 1) u_{1}^{'} = w_{1}, \\ (2 u_{1} + 1) w_{1}^{'} = - (2 u_{1} + 1) u_{1} + ε (2 u_{1} + 1) \frac{sin t + λ u_{2} + γ_{1} w_{1} + ε λ (χ sin (t + ϖ) + λ u_{1} + γ_{2} w_{2})}{ε^{2} λ^{2} - 1}, \\ (2 u_{2} + 1) u_{2}^{'} = w_{2}, \\ (2 u_{2} + 1) w_{2}^{'} = - (2 u_{2} + 1) u_{2} + ε (2 u_{2} + 1) \frac{χ sin (t + ϖ) + λ u_{1} + γ_{2} w_{2} + ε λ (sin t + λ u_{2} + γ_{1} w_{1})}{ε^{2} λ^{2} - 1}, \end{cases}

(54)

with (uncoupled) unperturbed equation for

ε = 0

(see (8)):

{\begin{cases} (2 u_{1} + 1) u_{1}^{'} = w_{1}, \\ (2 u_{1} + 1) w_{1}^{'} = - (2 u_{1} + 1) u_{1}, \\ (2 u_{2} + 1) u_{2}^{'} = w_{2}, \\ (2 u_{2} + 1) w_{2}^{'} = - (2 u_{2} + 1) u_{2} . \end{cases}

(55)

Neglecting left multiplicators

2 u + 1

(

u = u_{1}, u_{2}

) in (55), we obtain the following system (see condition (C2)):

{\begin{cases} u_{1}^{'} = w_{1}, \\ w_{1}^{'} = - (2 u_{1} + 1) u_{1}, \\ u_{2}^{'} = w_{2}, \\ w_{2}^{'} = - (2 u_{2} + 1) u_{2} . \end{cases}

(56)

Clearly, condition (C1) is satisfied with

x_{0} = (- \frac{1}{2}, 0)

and

F (x) = {(w, - (2 u + 1) u)}^{*}, ω (x) = 2 u + 1, x = {(u, w)}^{*} .

The equation

u^{″} + (2 u + 1) u = 0

has the prime integral

u^{' 2} + \frac{4}{3} u^{3} + u^{2}

. A solution

u_{0} (s)

satisfying

{lim}_{s \to \infty} u_{0} (s) = - \frac{1}{2}

has to satisfy

u^{' 2} + \frac{4}{3} u^{3} + u^{2} - \frac{1}{12} = 0 \Leftrightarrow 3 u^{' 2} + (4 u - 1) {(u + \frac{1}{2})}^{2} = 0

with the solution

u_{0} (s) = \frac{1}{4} (1 - 3 {tanh}^{2} \frac{s}{2})

bounded on ℝ. Hence

γ (s) = {(\frac{1}{4} (1 - 3 {tanh}^{2} \frac{s}{2}), - 6 {csch}^{3} s {sinh}^{4} \frac{s}{2})}^{*} .

Note

ω (γ (s)) = 2 u_{0} (s) + 1 = \frac{3}{2} {sech}^{2} \frac{s}{2} > 0

. From

F^{'} (x_{0}) = (\begin{array}{c} 0 & 1 \\ 1 & 0 \end{array}),

we get

μ_{\pm} = \pm 1

and

γ_{\pm} = {(1, \pm 1)}^{*}

. Since

\nabla ω (x_{0}) = {(2, 0)}^{*}

, we derive

〈 \nabla ω (x_{0}), γ_{\pm} 〉 = 2 > 0

, and condition (C3) holds as well. Next, we have

\begin{matrix} G_{1} (x_{1}, x_{2}, t, ε, κ) = \frac{2 u_{1} + 1}{ε^{2} λ^{2} - 1} (\begin{array}{c} 0 \\ sin t + λ u_{2} + γ_{1} w_{1} + ε λ (χ sin (t + ϖ) + λ u_{1} + γ_{2} w_{2}) \end{array}), \\ G_{2} (x_{1}, x_{2}, t, ε, κ) = \frac{2 u_{2} + 1}{ε^{2} λ^{2} - 1} (\begin{array}{c} 0 \\ χ sin (t + ϖ) + λ u_{1} + γ_{2} w_{2} + ε λ (sin t + λ u_{2} + γ_{1} w_{1}) \end{array}); \end{matrix}

hence

G_{i} (x_{0}, x_{0}, t, ε, κ) = 0

and assumption (C2) is also verified. Here

x_{i} = {(u_{i}, w_{i})}^{*}

,

i = 1, 2

. Furthermore by (15)

θ (s) = \int_{0}^{s} ω (γ (τ)) d τ = \int_{0}^{s} \frac{3}{2} {sech}^{2} \frac{τ}{2} d τ = 3 tanh \frac{s}{2},

so

T_{\pm} = \pm 3

and

x_{h} (t) = γ (θ^{- 1} (t)) = \frac{1}{4} {(1 - \frac{t^{2}}{3}, t (\frac{t^{2}}{9} - 1))}^{*} .

Thus

\begin{matrix} ω (x_{h} (t)) = \frac{1}{6} (9 - t^{2}), \nabla ω (x_{h} (t)) = {(2, 0)}^{*}, \\ \frac{F (x_{h} (t))}{ω (x_{h} (t))} = \frac{1}{12} {(- 2 t, t^{2} - 3)}^{*}, F^{'} (x_{h} (t)) = (\begin{array}{c} 0 & 1 \\ \frac{t^{2}}{3} - 2 & 0 \end{array}) . \end{matrix}

So now (16) has the form

\begin{aligned} z_{1}^{'} = \frac{2 t}{9 - t^{2}} z_{1} + \frac{6}{9 - t^{2}} z_{2}, \\ z_{2}^{'} = - z_{1}, \end{aligned}

(57)

which has the solution

x_{h}^{'} (t) = (- \frac{t}{6}, \frac{1}{12} (t^{2} - 3))

. In other words, we deal with

z_{2}^{″} = \frac{2 t}{9 - t^{2}} z_{2}^{'} - \frac{6}{9 - t^{2}} z_{2},

(58)

possessing the solution

\frac{1}{12} (t^{2} - 3)

. Following [[27], p.327], the second solution of (58) is given by

\frac{1}{9} ((t^{2} - 3) arctanh \frac{t}{3} - 3 t) .

Consequently a fundamental matrix solution of (57) has the form

Z (t) = (\begin{array}{c} - \frac{t}{6} & \frac{2}{9} (\frac{3 (t^{2} - 6)}{t^{2} - 9} - t arctanh \frac{t}{3}) \\ \frac{1}{12} (t^{2} - 3) & \frac{1}{9} (t^{2} - 3) arctanh \frac{t}{3} - \frac{t}{3} \end{array}) .

Note

det Z (t) = \frac{1}{9 - t^{2}}

. The adjoint system of (57) is (see (45))

\begin{aligned} ζ_{1}^{'} = \frac{2 t}{t^{2} - 9} ζ_{1} + ζ_{2}, \\ ζ_{2}^{'} = \frac{6}{t^{2} - 9} ζ_{1} \end{aligned}

(59)

with the fundamental matrix solution

Z^{- 1 *} (t) = (\begin{array}{c} \frac{1}{9} (9 - t^{2}) ((t^{2} - 3) arctanh \frac{t}{3} - 3 t) & \frac{1}{12} (t^{2} - 9) (t^{2} - 3) \\ \frac{2}{3} (t^{2} - 6) - \frac{2}{9} t (t^{2} - 9) arctanh \frac{t}{3} & \frac{1}{6} t (t^{2} - 9) \end{array}) .

Note

{lim}_{t \to \pm 3} Z^{- 1 *} (t) = (\begin{array}{c} 0 & 0 \\ 2 & 0 \end{array})

. Thus we take

v_{1} (t) = (\begin{array}{c} \frac{1}{9} (t^{2} - 9) (3 t - (t^{2} - 3) arctanh \frac{t}{3}) \\ \frac{2}{3} (t^{2} - 6) - \frac{2}{9} t (t^{2} - 9) arctanh \frac{t}{3} \end{array})

and

v_{2} (t) = (\begin{array}{c} \frac{1}{12} (t^{2} - 9) (t^{2} - 3) \\ \frac{1}{6} t (t^{2} - 9) \end{array}) .

We now compute

M (α, κ)

. We have (see the appendix)

\begin{aligned} \int_{T_{-}}^{T_{+}} v_{1}^{*} (t) \frac{G_{1} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \\ = \frac{1}{3} (9 λ + 4 sin α (Γ (2 cos 3 - 3 sin 3) + cos 3 (- 9 - 2 Ci 6 + log 36 - 3 Si 6) \\ + sin 3 (3 + 3 Ci 6 - 3 log 6 - 2 Si 6))) = a_{11} λ + a_{12} sin α, \end{aligned}

(60)

where

Si (t) = \int_{0}^{t} \frac{sin s}{s} d s

is the sine integral function,

Ci (t) = - \int_{t}^{\infty} \frac{cos s}{s} d s

is the cosine integral function [[28], p.886] and Γ is the Euler constant. Similarly

\begin{aligned} \int_{T_{-}}^{T_{+}} v_{2}^{*} (t) \frac{G_{1} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \\ = - \frac{54 γ_{1}}{35} - 2 cos α (3 cos 3 + 2 sin 3) = a_{21} γ_{1} + a_{22} cos α \end{aligned}

(61)

and

\begin{aligned} \int_{T_{-}}^{T_{+}} v_{1}^{*} (t) \frac{G_{2} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t = a_{11} λ + a_{12} χ sin (α + ϖ), \\ \int_{T_{-}}^{T_{+}} v_{2}^{*} (t) \frac{G_{2} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t = a_{21} γ_{2} + a_{22} χ cos (α + ϖ) . \end{aligned}

(62)

Note

a_{11} = 3, a_{12} ≐ 9.7406, a_{21} = - \frac{54}{35}, a_{22} ≐ 5.37547 .

Consequently, the Melnikov function is now

M (α, κ) = (\begin{array}{c} a_{11} λ + a_{12} sin α \\ a_{21} γ_{1} + a_{22} cos α \\ a_{11} λ + a_{12} χ sin (α + ϖ) \\ a_{21} γ_{2} + a_{22} χ cos (α + ϖ) \end{array}) .

(63)

The equation

M (α, κ) = 0

is equivalent to

\begin{aligned} λ = - \frac{a_{12}}{a_{11}} sin α = - \frac{a_{12}}{a_{11}} χ sin (α + ϖ), \\ γ_{1} = - \frac{a_{22}}{a_{21}} cos α, γ_{2} = - \frac{a_{22}}{a_{21}} χ cos (α + ϖ) . \end{aligned}

(64)

So having

χ > 0

and

ϖ > 0

such that the equation

sin α - χ sin (α + ϖ) = 0

(65)

has a simple zero

α_{0}

with

sin α_{0} < 0

,

cos α_{0} > 0

and

cos (α_{0} + ϖ) > 0

, formulas (64) give a simple zero

(α_{0}, γ_{1, 0}, γ_{2, 0}, λ_{0})

of (63) with positive

γ_{1, 0}

,

γ_{2, 0}

,

λ_{0}

, and Corollary 4.5 can be applied to (51). If

χ cos ϖ \neq 1

, then (65) is equivalent to

tan α = \frac{χ sin ϖ}{1 - χ cos ϖ} .

(66)

Hence assuming

ϖ \in] π, 2 π [

and

χ cos ϖ < 1

, the right hand side of (66) is negative, and then

α_{0} = arctan \frac{χ sin ϖ}{1 - χ cos ϖ} \in] - \frac{π}{2}, 0 [

(67)

satisfies

sin α_{0} < 0

and

cos α_{0} > 0

. Since

α_{0}

satisfies (65) and

sin α_{0} < 0

, condition

cos (α_{0} + ϖ) > 0

is equivalent to

tan (α_{0} + ϖ) < 0

. Then using (66), we derive

0 > tan (α_{0} + ϖ) = \frac{sin ϖ}{cos ϖ - χ}, ϖ \neq \frac{3 π}{2} .

(68)

When $cos ϖ < 0$ , then (68) is not satisfied, since $sin ϖ < 0$ . So we take $ϖ \in] \frac{3 π}{2}, 2 π [$ and (68) gives also $χ < cos ϖ$ . Clearly $χ < cos ϖ$ implies $1 > χ cos ϖ$ .

Summarizing we see that for any fixed χ and ϖ satisfying

0 < χ < cos ϖ, ϖ \in] \frac{3 π}{2}, 2 π [,

(69)

the Melnikov function (63) has a simple zero

(α_{0}, γ_{1, 0}, γ_{2, 0}, λ_{0})

given by (64) and (67), and

γ_{1, 0} > 0

,

γ_{2, 0} > 0

,

λ_{0} > 0

. Hence in the region given by (69) we apply Corollary 4.5 to (51) with parameters

γ_{1}

,

γ_{2}

, λ near

γ_{1, 0}

,

γ_{2, 0}

,

λ_{0}

determined by (64) and (67), i.e.,

\begin{aligned} λ_{0} = - \frac{a_{12} χ sin ϖ}{a_{11} \sqrt{1 + χ^{2} - 2 χ cos ϖ}}, \\ γ_{1, 0} = - \frac{a_{22} (1 - χ cos ϖ)}{a_{21} \sqrt{1 + χ^{2} - 2 χ cos ϖ}}, \\ γ_{2, 0} = - χ \frac{a_{22} (cos ϖ - χ)}{a_{21} \sqrt{1 + χ^{2} - 2 χ cos ϖ}} \end{aligned}

(70)

for any fixed ϖ and χ satisfying (69). Summarizing, we get the following.

Theorem 5.1 For any fixed ϖ, χ satisfying (69) and then

α_{0}

,

γ_{1, 0}

,

γ_{2, 0}

,

λ_{0}

given by (67) and (70), there is an

ε_{0} > 0

and smooth functions

α, γ_{1}, γ_{2}, λ :] - ε_{0}, ε [\to R

with

α (0) = α_{0}

,

γ_{1} (0) = γ_{1, 0}

,

γ_{2} (0) = γ_{2, 0}

,

λ (0) = λ_{0}

, such that for any

ε \in] - ε_{0}, ε_{0} [∖ {0}

, system (51) with

γ_{1} = γ_{1} (ε)

,

γ_{2} = γ_{2} (ε)

,

λ = λ (ε)

, possesses a unique solution

(u_{1} (ε, t), u_{2} (ε, t))

on

] - 3 + α (ε), 3 + α (ε) [

such that

\begin{aligned} lim_{ε \to 0} sup_{t \in] - 3, 3 [} | u_{j} (ε, t + α (ε)) - \frac{1}{4} (1 - \frac{t^{2}}{3}) | {(9 - t^{2})}^{- 1} = 0, \\ lim_{ε \to 0} sup_{t \in] - 3, 3 [} | u_{j}^{'} (ε, t + α (ε)) + \frac{t}{6} | = 0 . \end{aligned}

(71)

Proof We apply Corollary 4.5 to (53). Since, in this case,

φ (θ^{- 1} (t)) = \frac{1}{2} \frac{9 - t^{2}}{9 + t^{2}}

, according to Corollary 4.5 (53) has a solution

(u_{j} (t), w_{j} (t))

,

j = 1, 2

, such that

\begin{array}{r} lim_{ε \to 0} sup_{t \in] - 3, 3 [} | u_{j} (ε, t + α (ε)) - \frac{1}{4} (1 - \frac{t^{2}}{3}) | {(9 - t^{2})}^{- 1} = 0, \\ lim_{ε \to 0} sup_{t \in] - 3, 3 [} | w_{j} (ε, t + α (ε)) - \frac{t}{4} (\frac{t^{2}}{9} - 1) | {(9 - t^{2})}^{- 1} = 0 . \end{array}

(72)

Now,

w_{j} (ε, t) = (1 + 2 u_{j} (ε, t)) u_{j}^{'} (ε, t)

and

\frac{t}{4} (\frac{t^{2}}{9} - 1) = (1 + 2 u_{h} (t)) u_{h}^{'} (t)

, where

u_{h} (t) = \frac{1}{4} (1 - \frac{t^{2}}{3})

. So, taking

t^{'} : = t + α (ε)

,

\begin{aligned} w_{j} (ε, t^{'}) - \frac{t}{4} (\frac{t^{2}}{9} - 1) \\ = (1 + 2 u_{j} (ε, t^{'})) u_{j}^{'} (ε, t^{'}) - (1 + 2 u_{h} (t)) u_{h}^{'} (t) \\ = (1 + 2 u_{j} (ε, t^{'})) (u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t)) + 2 (u_{j} (ε, t^{'}) - u_{h} (t)) u_{h}^{'} (t) \\ = (1 + 2 u_{h} (t)) (u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t)) + 2 (u_{j} (ε, t^{'}) - u_{h} (t)) (u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t)) \\ + 2 (u_{j} (ε, t^{'}) - u_{h} (t)) u_{h}^{'} (t) \\ = [\frac{9 - t^{2}}{6} + 2 (u_{j} (ε, t^{'}) - u_{h} (t))] (u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t)) \\ - \frac{t}{3} (u_{j} (ε, t^{'}) - u_{h} (t)) \end{aligned}

and then

\begin{aligned} [w_{j} (ε, t^{'}) - \frac{t}{4} (\frac{t^{2}}{9} - 1)] {(9 - t^{2})}^{- 1} \\ = [\frac{1}{6} + 2 \frac{u_{j} (ε, t^{'}) - u_{h} (t)}{9 - t^{2}}] (u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t)) - \frac{t}{3} \frac{u_{j} (ε, t^{'}) - u_{h} (t)}{9 - t^{2}}, \end{aligned}

(73)

or, equivalently,

\begin{aligned} [\frac{1}{6} + 2 \frac{u_{j} (ε, t^{'}) - u_{h} (t)}{9 - t^{2}}] (u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t)) \\ = [w_{j} (ε, t^{'}) - \frac{t}{4} (\frac{t^{2}}{9} - 1)] {(9 - t^{2})}^{- 1} + \frac{t}{3} \frac{u_{j} (ε, t^{'}) - u_{h} (t)}{9 - t^{2}} . \end{aligned}

(74)

Since

{sup}_{- 3 < t < 3} | u_{j} (ε, t^{'}) - u_{h} (t) | {(9 - t^{2})}^{- 1} = O (ε)

we see that, for ε sufficiently small,

\frac{1}{7} < | \frac{1}{6} + 2 \frac{u_{j} (ε, t^{'}) - u_{h} (t)}{9 - t^{2}} | < \frac{1}{5},

and hence, using (74) and (72),

lim_{ε \to 0} sup_{- 3 < t < 3} | u_{j}^{'} (ε, t^{'}) - u_{h}^{'} (t) | = 0 .

(75)

Vice versa, if (75) and the first of (72) hold, then (73) gives

lim_{ε \to 0} sup_{- 3 < t < 3} | w_{j} (ε, t + α (ε)) - \frac{t}{4} (\frac{t^{2}}{9} - 1) | {(9 - t^{2})}^{- 1} = 0 .

Hence (71) and (72) are equivalent. The proof is complete. □

Of course, solutions given by Theorem 5.1 vary smoothly with respect $(ϖ, χ)$ satisfying (69).

Remark 5.2 Missed in the above analysis is the second possibility when

ϖ \in] 0, π [

. Then

sin ϖ > 0

and (66) is negative if

κ cos ϖ > 1,

(76)

so we get

ϖ \in] 0, \frac{π}{2} [

and

κ > 1

. Then inequality of (68) is satisfied since

κ > 1 > cos ϖ

. So we conclude that the result of Theorem 5.1 is valid also for

\begin{matrix} κ cos ϖ > 1, ϖ \in] 0, \frac{π}{2} [, \\ λ_{0} = \frac{a_{12} χ sin ϖ}{a_{11} \sqrt{1 + χ^{2} - 2 χ cos ϖ}}, \\ γ_{1, 0} = \frac{a_{22} (1 - χ cos ϖ)}{a_{21} \sqrt{1 + χ^{2} - 2 χ cos ϖ}}, \\ γ_{2, 0} = χ \frac{a_{22} (cos ϖ - χ)}{a_{21} \sqrt{1 + χ^{2} - 2 χ cos ϖ}} . \end{matrix}

Appendix

Let

v_{12} (t) = \frac{2}{3} (t^{2} - 6) - \frac{2}{9} t (t^{2} - 9) arctanh \frac{t}{3}

be the second component of

v_{1} (t)

. Note that

v_{12} (t)

is an even function and then

\begin{aligned} \int_{T_{-}}^{T_{+}} v_{1}^{*} (t) \frac{G_{1} (x_{h} (t), x_{h} (t), t + α, 0, κ)}{ω (x_{h} (t))} d t \\ = - \int_{- 3}^{3} v_{12} (t) [γ_{1} \frac{t}{4} (\frac{t^{2}}{9} - 1) + \frac{λ}{4} (1 - \frac{t^{2}}{3}) + sin t cos α + sin α cos t] d t \\ = - \int_{- 3}^{3} v_{12} (t) [\frac{λ}{4} (1 - \frac{t^{2}}{3}) + sin α cos t] d t \\ = \frac{λ}{12} \int_{- 3}^{3} v_{12} (t) (t^{2} - 3) d t - \int_{- 3}^{3} v_{12} (t) cos t d t sin α . \end{aligned}

(77)

Similarly,

\begin{aligned} \int_{T_{-}}^{T_{+}} v_{1}^{*} (t) G_{2} (x_{h} (t), x_{h} (t), t + α, \end{aligned}