A bifurcation problem for a class of periodically perturbed autonomous parabolic equations
© Kamenskii et al.; licensee Springer. 2013
Received: 24 December 2012
Accepted: 6 April 2013
Published: 23 April 2013
The paper deals with the problem of the existence of a branch of T-periodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear T-periodic term of small amplitude.
We solve this problem by first introducing a novel integral operator, whose fixed points are T-periodic solutions of the considered equation and vice versa. Then we compute the Malkin bifurcation function associated to this integral operator and we provide conditions under which the well-known assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.
MSC:35K58, 35B10, 35B20, 35B32.
Keywordsautonomous parabolic equations periodic perturbations limit cycle bifurcation periodic solutions
where , , ψ is T-periodic with respect to time and is a small parameter. Precisely, one seeks for the existence of a family of T-periodic solutions originating from a limit cycle of the autonomous unperturbed system.
It is assumed that the linearized system has only one characteristic multiplier with absolute value 1.
Since the pioneering papers [8, 9], a relevant bibliography devoted to this subject has been developed. From this bibliography, we quote in the sequel some of the papers more related to the present paper. In , the case when the cycle is not isolated was considered. By means of suitably defined bifurcation functions , , called Melnikov subharmonic functions, the existence of periodic solutions near to was proved. The periods of the solutions are in ratio with respect to the period of the perturbation term. The case when 1 is not a simple multiplier of the linearized system was treated in . The existence of at least two branches of T-periodic solutions originating from is shown in [12, 13] and their stability, in the sense of Lyapunov, follows from the results of [14, 15]. Developments of the Malkin’s and Melnikov’s approaches have permitted to prove several results about the existence of bifurcating solutions in [16–20]. Furthermore, the use of a Melnikov function permits to detect chaotic behavior of a suitable iterate of the Poincaré map associated to the differential equation (1), which is assumed to have a homoclinic orbit. Indeed, the existence of a simple zero of the considered Melnikov function ensures such a chaotic behavior; see [21, 22].
where and are given by and with singular . Then, by a convenient scaling of the variable x, we introduce an equivalent equation . For this equation, under the usual assumption of the existence of a simple zero of the Malkin bifurcation function associated to (1), the classical implicit function theorem can be applied to prove the existence of a branch of solutions originating from .
The same approach has been employed in  for a class of systems for which the resulting operators P and Q satisfy regularity conditions, which permit to apply the implicit function theorem, only along certain directions at the point of the limit cycle. Conditions to ensure the existence of several branches of T-periodic solution emanating from are provided by means of suitably defined Malkin bifurcation functions.
In all the papers cited before, the existence of periodic solutions for small is a consequence of the application of a convenient version of the implicit function theorem. This requires, as assumed for system (1), that and . Under less restrictive regularity conditions, by using topological tools such as the coincidence degree , the Leray-Schauder degree and the related continuation principles [26–28], existence results for small have been proved in  when the autonomous system is Hamiltonian and in [30, 31] when the limit cycle is isolated. More precisely, in [29, 31] the existence of two branches of T-periodic solutions was proved. Roughly speaking, in these papers, the bifurcation functions are employed to guarantee that the topological degree of certain operators is different from zero, rather than for the application of an implicit function theorem.
Topological degree arguments have been also employed in  to show the existence of periodic solutions for small in the case when the unperturbed system is nonautonomous and the perturbation consists of two nonlinear periodic terms with multiplicative different powers of . Finally, the behavior of the bifurcating periodic solutions when the perturbation vanishes has been studied in  for a nonsmooth system of the form (1) having an isolated limit cycle and in  for nonsmooth planar Hamiltonian systems.
where A is the infinitesimal generator of a strongly continuous semigroup , , acting in the Banach space E, satisfying the Radon-Nikodym property, is twice continuously Frechét differentiable and is continuously Frechét differentiable with respect to x, ε and T-periodic with respect to time. The functions ϕ and ψ satisfy suitable condensivity conditions with respect to the Hausdorff measure of noncompactness. The crucial assumption is that the unperturbed equation at has a continuous T-periodic isolated solution , i.e., .
has as simple zero. Here, for any , is given by , is the eigenvector corresponding to the simple eigenvalue 0 of and denotes the duality pairing of E with its dual . The main difficulty to verify the conditions of [, Theorem 2] consists in proving that the zero eigenvalue of is simple. In fact, in this case, the assumption that the linearized equation, around the limit cycle , , of the autonomous system at , does not have neither T-periodic solutions linearly independent with nor Floquet adjoint solutions to does not guarantee that the zero eigenvalue of is simple. To overcome this difficulty, we define in Section 3 a novel integral operator, equivalent to that associated to (2) with the property that for the resulting equation , the operator has 0 as simple eigenvalue.
Furthermore, in Section 4, we calculate the Malkin bifurcation function associated to the integral operator introduced in Section 3, and we formulate in Theorem 2 the result of the existence of a branch of T-periodic solutions parameterized by small. Proposition 1 of Section 5 states a somewhat surprising result: the Malkin functions associated to the two integral operators coincide and they have the common form of the classical Malkin function introduced for ordinary differential equations in finite dimensional spaces of the form (1). Finally, in Section 6, we provide a concrete example of a system of partial differential equations to which our abstract bifurcation result applies.
2 Assumptions and statement of the problem
from a T-periodic limit cycle of the unperturbed system corresponding to . Here, A is the infinitesimal generator of a strongly continuous semigroup , , acting in the Banach space E, which satisfies the Radon-Nikodym property; see [, Theorem 23, p.276]; , is T-periodic and , the space of T-periodic continuous functions .
Throughout the paper, we assume the following conditions on A, ϕ and ψ.
(H1) and there exists such that
(H2) ϕ is twice continuously Fréchet differentiable, ψ is continuously Fréchet differentiable with respect to the pair . Moreover, for any nonempty, bounded set we have
where , and is the Hausdorff measure of noncompactness .
(H3) The unperturbed equation(4)has a T-periodic isolated solution , hence the set of shifts , for any , represents a family of T-periodic solutions to (4). Moreover, for , we have that is a T-periodic solution to the linearized equation(5)where . We assume that for any and that (5) does not possess neither T-periodic solution linearly independent with nor Floquet adjoint solution to , whenever , i.e., (5) does not have solutions of the form
where is a T-periodic function.
We pose the following.
Problem To find conditions to ensure the existence of a branch of T-periodic solutions to (3) parameterized by , originating, for some , from the family of T-periodic solutions .
In conclusion, our problem will be solved if we show that for sufficiently small, equation (7) has a solution. To this end, it would be sufficient to verify the conditions of the following result.
Theorem 1 ([, Theorem 2])
Let B be a Banach space, let be a twice continuously Fréchet differentiable map and continuously Fréchet differentiable with respect to both the variables.
where is the eigenvector corresponding to the simple eigenvalue . Here, ∗ denotes the adjoint operator.
As it has been observed in , the compactness of the operator can be replaced by the condensivity of with respect to the Hausdorff measure of noncompactness. Indeed, as it is shown in , under assumptions (H1)-(H2), the operator and thus , see [, Theorem 1.5.4], are condensing with constant . Furthermore, [, Theorem 2.6.11] ensures that zero is an eigenvalue of of finite multiplicity. Therefore, under assumptions (H1)-(H3), one can easily verify that the conditions of the previous Theorem 1 are satisfied for (7) except the condition of the simplicity of the zero eigenvalue of , . In fact, the assumption that (5) does not possess neither T-periodic solutions linearly independent with , nor Floquet adjoint solutions to , whenever , does not imply that the zero eigenvalue of is simple, despite the existence of a bijection between the T-periodic solutions to (5) and the T-periodic solutions to . Moreover, as it shown in  the simplicity of the zero eigenvalue of does not imply that the T-periodic solution to (5) has the property mentioned above.
In conclusion, in order to apply Theorem 1, we will introduce a novel integral operator whose fixed points are also fixed points of (6) and vice versa, and thus T-periodic solutions to (3). Moreover, we will show that the zero eigenvalue of the corresponding operator is simple. This is the aim of the next section.
3 A novel equivalent integral operator
where is defined in the next lemma and is a given point.
For any fixed , let , and where .
We can now formulate the following result.
Lemma 1 Assume (H1)-(H3) and that and satisfy the conditions:
Since from (10), it follows that and so from (8) we get .
By our definition , , with and . Therefore, (19) is satisfied and this concludes the proof. □
Remark 1 Observe that a little though convinces of the existence of and satisfying the conditions (H4)-(H6) of Lemma 1.
Furthermore, recall that assumptions (H1)-(H2) ensure that the operator is condensing with constant (see ); moreover, [, Theorem 1.5.4] guarantees that is also condensing with the same constant. Finally, by [, Theorem 2.6.11], zero turns out to be an eigenvalue of of finite multiplicity. The second part of the proof of Lemma 1 shows that it is simple.
4 The Malkin bifurcation function
For notational convenience, we simply denote by . In order to compute , it is necessary to determine in explicit form. The following result solves the problem.
where γ is an eigenvector of corresponding to the eigenvalue 1 and is the characteristic function of the interval .
Lemmas 1 and 2, together with the fact that (H1)-(H2) ensure the condensivity of , , of constant (see ) allow to apply Theorem 1 to state the following.
for sufficiently small and .
5 An invariance property of the Malkin bifurcation function
In what follows, we state an interesting property of the Malkin bifurcation functions introduced before. Precisely, we can prove the following result.
Proposition 1 Let , assume that is simple. Then the Malkin bifurcation function associated to system (6) coincide with the Malkin bifurcation function associated to system (8).
6 An example
In order to introduce an example consistent with the general approach of the paper that requires the employ of the theory of condensing operators, we are led to consider partial differential equations of hyperbolic type, whose abstract formulation in Banach spaces gives rise to infinitesimal generators of noncompact -semigroups; see, e.g., .
where , , Ω is a -smooth bounded domain in , ν denotes the outward unit normal vector to Q, λ is a small positive parameter representing the thickness of the domain of the variable λy, , , , are positive constants and the functions , are of class jointly in their arguments. The linear part of system (26) generates an exponentially stable -semigroup in a suitable Banach space; see  as well as the related references therein. Under the assumption of the existence of a -periodic solution of the limit problem, obtained as , and suitable conditions on the growth of the derivatives of , with respect to their arguments, it is shown in  that [, Theorem 1] applies. This result guarantees the existence of such that, for fixed , system (26) has an isolated -periodic solution . The crucial assumption of [, Theorem 1] is that the zero eigenvalue of the linearized system around is simple. For a single damped wave equation of system (26) with the nonlinear term ϕ depending periodically on time t, the existence of periodic solutions was studied in .
If we assume that the superposition operators generated by the functions , , , satisfy assumption (H2) of this paper, then (H1) and [, Theorem 4.3.1] ensure that the -semigroup generated by the linearization around of the unperturbed system, corresponding to in (27), is strongly contractive with respect to the Hausdorff measure of noncompactness , i.e., χ-strongly contractive. Therefore, our abstract bifurcation result Theorem 2 applies to system (27).
Dedicated to professor Jean Mawhin on the occasion of his seventieth birthday.
The first two authors acknowledge the support by RFBR Grants 10-01-93112-a and 12-01-0392-a. The third one acknowledges the support by the GNAMPA of the Istituto di Alta Matematica. The authors would like also to thank the referees for their helpful comments and suggestions which improved the presentation of the paper.
- Awrejcewicz J, Lamarque C-H: Bifurcation and Chaos in Nonsmooth Mechanical System. World Scientific, Singapore; 2003.
- Awrejcewicz J, Holicke MM: Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods. World Scientific, Singapore; 2007.
- di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P: Piecewise-Smooth Dynamical Systems, Theory and Applications. Springer, London; 2008.
- Fecčkan M: Topological Degree Approach to Bifurcation Problems. Springer, Berlin; 2008.View Article
- Fecčkan M: Bifurcation and Chaos in Discontinuous and Continuous Systems. Springer, Berlin; 2011.View Article
- di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Olivar G, Piiroinen PT: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 2008, 50: 629-701. 10.1137/050625060MathSciNetView Article
- Mawhin J: Topological bifurcation theory: old and new. Nankai Ser. Pure Appl. Math. Theoret. Phys. 7. In Progress in Variational Methods. World Scientific, Hackensak; 2011:169-186.
- Loud WS: Periodic solutions of a perturbed autonomous system. Ann. Math. 1959, 70: 490-529. 10.2307/1970327MathSciNetView Article
- Malkin IG: On Poincaré’s theory of periodic solutions. Akad. Nauk SSSR. Prikl. Mat. Meh. 1949, 13: 633-646. (in Russian)MathSciNet
- Melnikov VK: On the stability of a center for time-periodic perturbations. Tr. Mosk. Mat. Obŝ. 1963, 12: 3-52. (in Russian)
- Rhouma MBH, Chicone C: On the continuation of periodic orbits. Methods Appl. Anal. 2000, 7: 85-104.MathSciNet
- Hausrath AR, Manásevich RF: Periodic solutions of periodically harvested Lotka-Volterra systems. Rev. Colomb. Mat. 1987, 21: 337-345.
- Hausrath AR, Manásevich RF: Periodic solutions of periodically forced nondegenerate systems. Rocky Mt. J. Math. 1988, 18: 49-65. 10.1216/RMJ-1988-18-1-49View Article
- Hausrath AR, Manásevich RF: Stability properties of periodic solutions of periodically forced non degenerate systems. Rocky Mt. J. Math. 1988, 18: 785-800. 10.1216/RMJ-1988-18-4-785View Article
- Ortega R: A criterion for asymptotic stability based on topological degree. Proceedings of the First World Congress of Nonlinear Analysts, Tampa, FL 1992, 383-394.
- Blekhman II: Synchronization in Nature and Engineering. Nauka, Moscow; 1981. (in Russian)
- Guckenheimer J, Holmes P Applied Mathematical Sciences 42. In Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York; 1983.View Article
- Greenspan B, Holmes P: Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators. SIAM J. Math. Anal. 1984, 15: 69-97. 10.1137/0515003MathSciNetView Article
- Lazer AC: Small periodic perturbations of a class of conservative systems. J. Differ. Equ. 1973, 13: 438-456. 10.1016/0022-0396(73)90003-XMathSciNetView Article
- Tkhai VN: Periodic motions of a system close to an autonomous reversible system. J. Appl. Math. Mech. 2001, 65: 647-664. 10.1016/S0021-8928(01)00069-7MathSciNetView Article
- Palmer KJ: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ. 1984, 55: 225-256. 10.1016/0022-0396(84)90082-2View Article
- Battelli F, Fecčkan M: Some remarks on the Melnikov function. Electron. J. Differ. Equ. 2002, 2002: 1-29.MathSciNet
- Kamenskii M, Makarenkov O, Nistri P: An alternative approach to study bifurcation from a limit cycle in periodically perturbed autonomous systems. J. Dyn. Differ. Equ. 2011, 23: 425-435. doi:10.1007/s10884-011-9207-4 10.1007/s10884-011-9207-4MathSciNetView Article
- Kamenskii M, Mikhaylenko B, Nistri P: Nonsmooth bifurcation in finite dimensional spaces via scaling of variables. Differ. Equ. Dyn. Syst. 2012, 20: 191-205. 10.1007/s12591-011-0102-6MathSciNetView Article
- Mawhin J CBMS Reg. Conf. Ser. Math. 40. In Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979.
- Mawhin J: Leray-Schauder continuation theorems in the absence of a priori bounds. Topol. Methods Nonlinear Anal. 1997, 9: 179-200.MathSciNet
- Mawhin J: Periodic solutions in the golden sixties: the birth of a continuation theorem. In Ten Mathematical Essays on Approximation in Analysis and Topology. Elsevier, Amsterdam; 2005:199-214.View Article
- Capietto A, Mawhin J, Zanolin F: Continuation theorems for periodic perturbations of autonomous systems. Trans. Am. Math. Soc. 1992, 329: 41-72. 10.1090/S0002-9947-1992-1042285-7MathSciNetView Article
- Henrard M, Zanolin F: Bifurcation from a periodic orbit in perturbed planar Hamiltonian systems. J. Math. Anal. Appl. 2003, 277: 79-103. 10.1016/S0022-247X(02)00504-8MathSciNetView Article
- Kamenskii M, Makarenkov O, Nistri P: A continuation principle for a class of periodically perturbed autonomous systems. Math. Nachr. 2008, 281: 42-61. 10.1002/mana.200610586MathSciNetView Article
- Makarenkov O, Nistri P: Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations. J. Math. Anal. Appl. 2008, 338: 42-61.MathSciNetView Article
- Kamenskii M, Makarenkov O, Nistri P: Small parameter perturbations of nonlinear periodic systems. Nonlinearity 2004, 17: 193-205. 10.1088/0951-7715/17/1/012MathSciNetView Article
- Makarenkov O, Nistri P: On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Commun. Pure Appl. Anal. 2008, 17: 49-61.MathSciNet
- Makarenkov O, Malaguti L, Nistri P: On the behavior of periodic solutions of planar autonomous Hamiltonian systems with multivalued periodic perturbations. Z. Anal. Anwend. 2011, 30: 129-144.MathSciNetView Article
- Couchouron JF, Kamenski M, Nistri P: An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type. Commun. Pure Appl. Anal. 2013, 12: 1845-1859. doi:10.3934/cpaa.2013.12.1845MathSciNetView Article
- Kamenskii MI, Mikhaylenko BA: Averaging principle and the variational approach in the problem on the bifurcation of periodic solutions from non-isolated equilibria of the averaged equation. Differ. Equ. 2012, 48: 1-8. 10.1134/S0012266112010016MathSciNetView Article
- Royden HL: Real Analysis. 3rd edition. Macmillan Co., New York; 1988.
- Akhmerov RR, Kamenski MI, Potapov AS, Rodkina AN, Sadovskii BN Operator Theory: Advances and Applications 55. In Measures of Noncompactness and Condensing Operators. Birkhäuser, Basel; 1992.View Article
- Kamenskii MI, Mikhaylenko BA: On perturbations of systems with multidimensional degeneration. Autom. Remote Control 2011, 72: 1036-1047. 10.1134/S0005117911050110MathSciNetView Article
- Kamenskii M, Nistri P:An averaging method for singularly perturbed systems of semi linear differential inclusions with -semigroups. Set-Valued Anal. 2003, 11: 345-357. 10.1023/A:1025604317289MathSciNetView Article
- Mikhaylenko BA: On equivalent integral operators in periodic solutions problems of differential equations. Proc. Voronezh State Univ. Voronezh 2011, 1: 193-201. (in Russian)
- Pazy A Applied Mathematical Sciences 44. In Semigroups of Linear Operator and Applications to Partial Differential Equations. Springer, New York; 1983.View Article
- Johnson R, Kamenskii M, Nistri P: Existence of periodic solutions for an autonomous damped wave equation in a thin domain. J. Dyn. Differ. Equ. 1998, 10: 409-424. 10.1023/A:1022601213052MathSciNetView Article
- Johnson R, Kamenskii M, Nistri P: Erratum to existence of periodic solutions for an autonomous damped wave equation in a thin domain. J. Dyn. Differ. Equ. 2000, 12: 675-679. 10.1023/A:1026419913334MathSciNetView Article
- Raugel G Lecture Notes in Mathematics 1609. In Dynamics of Partial Differential Equations in Thin Domains. Springer, Berlin; 1995:208-315.
- Johnson R, Kamenskii M, Nistri P: On periodic solutions of a damped wave equation in a thin domain using degree theoretic methods. J. Differ. Equ. 1997, 140: 186-208. 10.1006/jdeq.1997.3301MathSciNetView Article
- Kamenskii M, Obukhovskii V, Zecca P De Gruyter Series in Nonlinear Analysis and Applications. In Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter, Berlin; 2001.View Article
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.