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.
Keywords:autonomous parabolic equations; periodic perturbations; limit cycle; bifurcation; periodic solutions
In recent years, bifurcation problems for smooth and nonsmooth dynamical systems have received a renewed attention and interest from different fields of engineering, physics and mathematics. We mention here, among others, the monographs [1-5] and the review papers [6,7]. Of particular interest is the study of the bifurcation of periodic solutions for periodically perturbed autonomous systems of the form:
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.
Existence, uniqueness and asymptotic stability of bifurcating periodic solutions for system (1) are classical problems; see [8,9]. The main tool employed in these papers is the so-called Malkin bifurcation function:
where is a T-periodic solution of
the adjoint system of the linearized 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].
Very recently, in , a new method to prove bifurcation of a branch of asymptotically stable periodic solutions to (1) has been proposed. The method consists first in converting the problem of finding fixed points of the singular Poincaré map , , associated to (1) into the problem of the existence of zeros of an equation of the form:
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.
A first attempt to extend to infinite dimensional bifurcation problem the approach outlined in  has been presented in , with the aim of studying the bifurcation of periodic solutions for a functional differential equation of neutral type. In the present paper, we precise and generalize the idea of how to use a suitable abstract Malkin bifurcation function to deal with infinite dimensional bifurcation problems. To this aim, we consider the following autonomous differential equation of parabolic type periodically perturbed by a nonlinear term of small amplitude:
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., .
The paper is organized as follows. In Section 2, we precise the conditions under which there is at least a branch of T-periodic solutions to (2) emanating from . This existence result follows from the application of [, Theorem 2]; this theorem relies on the method introduced in . Precisely, to solve the bifurcation problem for (2), we introduce an equivalent integral equation whose zeros are the T-periodic solutions to (2) and that we rewrite in the following form:
where and . This equation has a branch of solutions originating from if the Malkin bifurcation function given by
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
The paper deals with the problem of the existence of bifurcation of T-periodic solutions for the T-periodically perturbed autonomous equation of the form
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
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
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 .
To solve this problem, we first reduce the existence of T-periodic solutions to (3) to the problem of finding fixed points of an integral equation. For this, we introduce the linear operator as follows:
Therefore, if we let
then a function satisfying
is a solution to (3) and vice versa. Moreover, it is easy to verify that the equation
is equivalent to the linearized unperturbed equation
where . Hence, we can rewrite (6) in the following form:
where and are defined as follows:
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])
LetBbe a Banach space, let be a twice continuously Fréchet differentiable map and continuously Fréchet differentiable with respect to both the variables.
Assume that the equation has one-dimensional set of solutions , parameterized by , such that there exists for any and for any . Assume that the eigenvalue is simple and the operator is compact, whenever . Consider the function defined by
where is the eigenvector corresponding to the simple eigenvalue . Here, ∗ denotes the adjoint operator.
Then, for each such that and the equation is solvable, for sufficiently small, in a neighborhood of the point and the solution has the form
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
Let , hence equation (7) reads as . Consider the integral equation
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 1Assume (H1)-(H3) and that and satisfy the conditions:
Define as follows:
where . Then (8) is equivalent to (6). Moreover, the zero eigenvalue of is simple, where is given by
Proof First of all observe that, under our assumptions, we have that
Indeed, arguing by contradiction assume that is such that
then, by the definition of ξ, we obtain
thus and are linearly dependent, i.e., is also an eigenvector of , that is,
In conclusion, we should have
On the other hand, as it is easy to verify our conditions imply that
We now prove the equivalence between (6) and (8). Clearly, if for some then (8) is satisfied. Conversely, assume that is a solution to (8), hence
Integrating on the interval , we obtain
Since from (10), it follows that and so from (8) we get .
It remains to prove the second part of the lemma. For this, given , let , , and let . To simplify the notation in the sequel, we omit the subscript θ. Observe that
Then the equation
can be rewritten as follows:
For , define
then the equation (11) takes the form
Clearly, is an eigenvector of corresponding to the zero eigenvalue, i.e., . Assume now that there exists an adjoint vector to , namely
for any . By assumption, there are no adjoint Floquet solutions to (5), thus
does not possess T-periodic solution . The integral form of (13) is given by
Therefore, it remains to show that (12) and (14) coincide, namely
For this, integrating (12) on the interval , we obtain
On the other hand , thus
From (16), we get the following form for (15)
Finally substituting (18) into (17), we obtain
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
In the previous section, Lemma 1 states that the operator P, associated to the integral equation (8) satisfies the conditions of Theorem 1. This section is devoted to the computation of the following Malkin bifurcation function associated to (8)
where is an eigenvector of , i.e.,
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.
Lemma 2Assume (H1)-(H6), we have that
whereγis an eigenvector of corresponding to the eigenvalue 1 and is the characteristic function of the interval .
Proof By assumption, E has the Radon-Nikodym property, then the eigenvalue of can be determined, without loss of generality in the Hilbert space . Hence, for any we have
First, by using assumption (H1), we calculate
Therefore, we have
We now calculate the adjoint operator for . For this, consider
Finally, we calculate , for this consider
Now, we are in the position to determine the eigenvector of . We have that
Let , then the previous equation takes the form
namely γ is an eigenvector of the linear operator corresponding to the simple eigenvalue 1. Therefore, by replacing y with γ in (20), we obtain
The aim now is to find γ, for this consider the adjoint equation to (5)
and the solution of (22), defined for , given by
For , we have
Since v is T-periodic, we obtain
By using (24) into (23), we get
Put , then and (25) becomes
i.e., is a solution to (21). Hence,
where is the T-periodic solution to the adjoint equation (22). Finally, from
By (9), we get
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.
Theorem 2Assume (H1)-(H6). If there exists such that and . Then there exists a branch ofT-periodic solutions to (3) of the form
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 1Let , assume that is simple. Then the Malkin bifurcation function associated to system (6) coincide with the Malkin bifurcation function associated to system (8).
Let , then by (9) we have
Remark 3 can be rewritten in the classical form of the Malkin bifurcation function for ordinary differential equations as (1) of the Introduction. In fact, consider
Since , , Ψ is the superposition operator generated by ψ and v solves (22) we have
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., .
Precisely, following  and , we present a concrete, not academic example concerning the existence of periodic solutions of a system of two autonomous damped wave equations in a thin domain with Neumann boundary conditions. The study of the dynamics of partial differential equations in thin domains has received many attention in the past few years; see  and the extensive references therein. The system has the form
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 .
Consider now a -periodic perturbation of (26) of small amplitude
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).
The authors declare that they have no competing interests.
The authors wrote this paper in collaboration and with the same responsibility. All authors read and approved the final version of the manuscript.
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.
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