A bifurcation problem for a class of periodically perturbed autonomous parabolic equations
 Mikhail Kamenskii^{1},
 Boris Mikhaylenko^{1} and
 Paolo Nistri^{2}Email author
DOI: 10.1186/168727702013101
© Kamenskii et al.; licensee Springer. 2013
Received: 24 December 2012
Accepted: 6 April 2013
Published: 23 April 2013
Abstract
The paper deals with the problem of the existence of a branch of Tperiodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear Tperiodic term of small amplitude.
We solve this problem by first introducing a novel integral operator, whose fixed points are Tperiodic 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 wellknown 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 solutions1 Introduction
where $\varphi \in {C}^{2}({\mathbb{R}}^{n},{\mathbb{R}}^{n})$, $\psi \in {C}^{1}(\mathbb{R}\times {\mathbb{R}}^{n}\times [0,1],{\mathbb{R}}^{n})$, ψ is Tperiodic with respect to time and $\epsilon \ge 0$ is a small parameter. Precisely, one seeks for the existence of a family of Tperiodic solutions originating from a limit cycle ${x}_{0}$ 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 [10], the case when the cycle ${x}_{0}$ is not isolated was considered. By means of suitably defined bifurcation functions ${f}_{m,n}$, $m,n\in \mathbb{N}$, called Melnikov subharmonic functions, the existence of periodic solutions near to ${x}_{0}$ was proved. The periods of the solutions are in $m:n$ 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 [11]. The existence of at least two branches of Tperiodic solutions originating from ${x}_{0}$ 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 ${\mathrm{\Pi}}_{\epsilon}$ 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 $P:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ and $Q:{\mathbb{R}}^{n}\times [0,1]\to {\mathbb{R}}^{n}$ are given by $P(x)={\mathrm{\Pi}}_{0}(x)x$ and $Q(x,\epsilon )=\frac{{\mathrm{\Pi}}_{\epsilon}(x){\mathrm{\Pi}}_{0}(x)}{\epsilon}$ with singular ${P}^{\prime}({x}_{0})$. Then, by a convenient scaling of the variable x, we introduce an equivalent equation $\mathrm{\Psi}(w,\epsilon )=0$. 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 ${x}_{0}$.
The same approach has been employed in [24] 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 ${x}_{0}(\cdot )$ of the limit cycle. Conditions to ensure the existence of several branches of Tperiodic solution emanating from ${x}_{0}$ are provided by means of suitably defined Malkin bifurcation functions.
In all the papers cited before, the existence of periodic solutions for $\epsilon \ge 0$ small is a consequence of the application of a convenient version of the implicit function theorem. This requires, as assumed for system (1), that $\varphi \in {C}^{2}({\mathbb{R}}^{n},{\mathbb{R}}^{n})$ and $\psi \in {C}^{1}(\mathbb{R}\times {\mathbb{R}}^{n}\times [0,1],{\mathbb{R}}^{n})$. Under less restrictive regularity conditions, by using topological tools such as the coincidence degree [25], the LeraySchauder degree and the related continuation principles [26–28], existence results for $\epsilon \ge 0$ small have been proved in [29] 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 Tperiodic 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 [32] to show the existence of periodic solutions for $\epsilon \ge 0$ small in the case when the unperturbed system is nonautonomous and the perturbation consists of two nonlinear periodic terms with multiplicative different powers of $\epsilon \ge 0$. Finally, the behavior of the bifurcating periodic solutions when the perturbation vanishes has been studied in [33] for a nonsmooth system of the form (1) having an isolated limit cycle and in [34] for nonsmooth planar Hamiltonian systems.
where A is the infinitesimal generator of a strongly continuous semigroup ${e}^{At}$, $t\ge 0$, acting in the Banach space E, satisfying the RadonNikodym property, $\varphi :E\to E$ is twice continuously Frechét differentiable and $\psi :\mathbb{R}\times E\times [0,1]\to E$ is continuously Frechét differentiable with respect to x, ε and Tperiodic 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 $\epsilon =0$ has a continuous Tperiodic isolated solution ${x}_{0}:\mathbb{R}\to E$, i.e., ${x}_{0}\in {C}_{T}(E)$.
has ${\theta}_{0}\in [0,T]$ as simple zero. Here, for any $\theta \in [0,T]$, ${x}_{0}(\theta )\in {C}_{T}(E)$ is given by ${x}_{0}(\theta )(\cdot ):={x}_{\theta}(\cdot )=x(\cdot +\theta )$, ${z}_{0}(\theta )$ is the eigenvector corresponding to the simple eigenvalue 0 of ${({\tilde{P}}^{\prime}({x}_{0}(\theta )))}^{\ast}$ and $\u3008\cdot ,\cdot \u3009$ denotes the duality pairing of E with its dual ${E}^{\prime}$. The main difficulty to verify the conditions of [[36], Theorem 2] consists in proving that the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ is simple. In fact, in this case, the assumption that the linearized equation, around the limit cycle ${x}_{0}(\theta )$, $\theta \in [0,T]$, of the autonomous system at $\epsilon =0$, does not have neither Tperiodic solutions linearly independent with ${x}_{0}^{\prime}(\theta )$ nor Floquet adjoint solutions to ${x}_{0}^{\prime}(\theta )$ does not guarantee that the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ 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 $P(x)+\epsilon Q(x,\epsilon )=0$, the operator ${P}^{\prime}({x}_{0}(\theta ))$ 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 Tperiodic solutions parameterized by $\epsilon \ge 0$ 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 Tperiodic limit cycle ${x}_{0}$ of the unperturbed system corresponding to $\epsilon =0$. Here, A is the infinitesimal generator of a strongly continuous semigroup ${e}^{At}$, $t\ge 0$, acting in the Banach space E, which satisfies the RadonNikodym property; see [[37], Theorem 23, p.276]; $\varphi :E\to E$, $\psi :\mathbb{R}\times E\times [0,1]\to E$ is Tperiodic and $x\in {C}_{T}(E)$, the space of Tperiodic continuous functions $x:\mathbb{R}\to E$.
Throughout the paper, we assume the following conditions on A, ϕ and ψ.

(H1) ${({e}^{At})}^{\ast}={e}^{{A}^{\ast}t}$ and there exists ${\alpha}_{0}>0$ such that${\parallel {e}^{At}\parallel}_{E}\le {e}^{{\alpha}_{0}t},\phantom{\rule{1em}{0ex}}t\ge 0.$

(H2) ϕ is twice continuously Fréchet differentiable, ψ is continuously Fréchet differentiable with respect to the pair $(x,\epsilon )$. Moreover, for any nonempty, bounded set $\mathrm{\Omega}\subset E$ we have$\begin{array}{c}\chi (\varphi (\mathrm{\Omega}))\le k\chi (\mathrm{\Omega});\hfill \\ \chi \left(\psi ([0,T]\times \mathrm{\Omega}\times [0,1])\right)\le l\chi (\mathrm{\Omega}),\hfill \end{array}$
where $0<k/{\alpha}_{0}<1$, $l>0$ and $\chi (\cdot )$ is the Hausdorff measure of noncompactness [38].

(H3) The unperturbed equation$\dot{x}=Ax+\varphi (x)$(4)has a Tperiodic isolated solution ${x}_{0}\in {C}_{T}(E)$, hence the set of shifts ${x}_{\theta}(\cdot )={x}_{0}(\cdot +\theta )$, for any $\theta \in [0,T]$, represents a family of Tperiodic solutions to (4). Moreover, for $\theta \in [0,T]$, we have that ${y}_{\theta}(t):={x}_{\theta}^{\prime}(t)$ is a Tperiodic solution to the linearized equation$\dot{y}=Ay+{\varphi}^{\prime}({x}_{0}(\theta ))y,$(5)where ${x}_{0}(\theta ):={x}_{\theta}$. We assume that ${x}_{0}^{\prime}(\theta )\ne 0$ for any $\theta \in [0,T]$ and that (5) does not possess neither Tperiodic solution linearly independent with ${y}_{\theta}(t)$ nor Floquet adjoint solution to ${y}_{\theta}(t)$, whenever $\theta \in [0,T]$, i.e., (5) does not have solutions of the form$y(t)=v(t)+\frac{t}{T}{y}_{\theta}(t),$
where $v(t)$ is a Tperiodic function.
We pose the following.
Problem To find conditions to ensure the existence of a branch of Tperiodic solutions to (3) parameterized by $\epsilon \ge 0$, originating, for some ${\theta}_{0}\in [0,T]$, from the family of Tperiodic solutions ${x}_{0}(\theta )$.
In conclusion, our problem will be solved if we show that for $\epsilon \ge 0$ sufficiently small, equation (7) has a solution. To this end, it would be sufficient to verify the conditions of the following result.
Theorem 1 ([[36], Theorem 2])
Let B be a Banach space, let $P:B\to B$ be a twice continuously Fréchet differentiable map and $Q:B\times [0,1]\to B$ continuously Fréchet differentiable with respect to both the variables.
where ${z}_{0}(\theta )$ is the eigenvector corresponding to the simple eigenvalue $0\in \sigma {({P}^{\prime}({x}_{0}(\theta )))}^{\ast}$. Here, ∗ denotes the adjoint operator.
where $w\in B$ can be determined in explicit form as shown in [[35], Theorem 2.3] and [[36], Lemma 3].
As it has been observed in [39], the compactness of the operator $I{P}^{\prime}({x}_{0}(\theta ))$ can be replaced by the condensivity of $I{P}^{\prime}({x}_{0}(\theta ))$ with respect to the Hausdorff measure of noncompactness. Indeed, as it is shown in [40], under assumptions (H1)(H2), the operator $IP$ and thus $I{P}^{\prime}({x}_{0}(\theta ))$, see [[38], Theorem 1.5.4], are condensing with constant $k/{\alpha}_{0}<1$. Furthermore, [[38], Theorem 2.6.11] ensures that zero is an eigenvalue of ${P}^{\prime}({x}_{0}(\theta ))$ 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 ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$, $\theta \in [0,T]$. In fact, the assumption that (5) does not possess neither Tperiodic solutions linearly independent with ${y}_{\theta}(t)$, nor Floquet adjoint solutions to ${y}_{\theta}(t)$, whenever $\theta \in [0,T]$, does not imply that the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ is simple, despite the existence of a bijection between the Tperiodic solutions to (5) and the Tperiodic solutions to ${\tilde{P}}^{\prime}({x}_{0}(\theta ))=0$. Moreover, as it shown in [41] the simplicity of the zero eigenvalue of ${\tilde{P}}^{\prime}({x}_{0}(\theta ))$ does not imply that the Tperiodic solution ${y}_{\theta}(t)$ 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 Tperiodic solutions to (3). Moreover, we will show that the zero eigenvalue of the corresponding operator ${P}^{\prime}({x}_{0}(\theta ))$ is simple. This is the aim of the next section.
3 A novel equivalent integral operator
where $\xi :E\to {C}_{T}(E)$ is defined in the next lemma and $\tau \in [0,T]$ is a given point.
For any fixed $\theta \in [0,T]$, let ${e}_{0}:={x}_{0}^{\prime}(\theta )\ne 0$, ${\gamma}_{0}:={\int}_{0}^{\tau}{e}_{0}(s)\phantom{\rule{0.2em}{0ex}}ds$ and $\beta :=\u3008f,{\int}_{0}^{\tau}(J{e}_{0})(s)\phantom{\rule{0.2em}{0ex}}ds\u3009$ where $f\in {E}^{\prime}$.
We can now formulate the following result.
Lemma 1 Assume (H1)(H3) and that $\tau \in [0,T]$ and $f\in {E}^{\prime}$ satisfy the conditions:

(H4) ${\gamma}_{0}\ne 0$.

(H5) $\beta \ne 0$.

(H6) $\u3008f,{\gamma}_{0}\u3009=1$.
Since $1\notin \sigma ({\int}_{0}^{\tau}\xi (s)\phantom{\rule{0.2em}{0ex}}ds)$ from (10), it follows that ${\int}_{0}^{\tau}{F}_{\epsilon}(y)(s)\phantom{\rule{0.2em}{0ex}}ds=0$ and so from (8) we get ${F}_{\epsilon}(y)=0$.
By our definition $\xi (t)x=\u3008f,x\u3009g(t)$, $g(t)=\frac{1}{\beta}({e}_{0}(t)+(J{e}_{0})(t))$, with $\u3008f,{\int}_{0}^{\tau}{e}_{0}(s)\phantom{\rule{0.2em}{0ex}}ds\u3009=1$ and $\u3008f,{\int}_{0}^{\tau}(J{e}_{0})(s)\phantom{\rule{0.2em}{0ex}}ds\u3009=\beta $. Therefore, (19) is satisfied and this concludes the proof. □
Remark 1 Observe that a little though convinces of the existence of $\tau \in [0,T]$ and $f\in {E}^{\prime}$ satisfying the conditions (H4)(H6) of Lemma 1.
Furthermore, recall that assumptions (H1)(H2) ensure that the operator $IP$ is condensing with constant $k/{\alpha}_{0}<1$ (see [40]); moreover, [[38], Theorem 1.5.4] guarantees that $I{P}^{\prime}({x}_{0}(\theta ))$ is also condensing with the same constant. Finally, by [[38], Theorem 2.6.11], zero turns out to be an eigenvalue of ${P}^{\prime}({x}_{0}(\theta ))$ 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 ${z}_{0}(\theta )\in {C}_{T}(E)$ by ${z}_{0}$. In order to compute ${M}_{\xi}(\theta )$, it is necessary to determine ${z}_{0}$ in explicit form. The following result solves the problem.
where γ is an eigenvector of ${(J\alpha )}^{\ast}$ corresponding to the eigenvalue 1 and ${\mathbb{1}}_{[0,\tau ]}$ is the characteristic function of the interval $[0,\tau ]$.
□
Lemmas 1 and 2, together with the fact that (H1)(H2) ensure the condensivity of $I{P}^{\prime}({x}_{0}(\theta ))$, $\theta \in [0,T]$, of constant $0<k/{\alpha}_{0}<1$ (see [40]) allow to apply Theorem 1 to state the following.
for $\epsilon \ge 0$ sufficiently small and $w\in {C}_{T}(E)$.
Remark 2 The function w can be calculated in an explicit form as shown in [[35], Theorem 2.3] and [[36], Lemma 3].
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 $\theta \in [0,T]$, assume that $0\in \sigma ({\tilde{P}}^{\prime}({x}_{0}(\theta )))$ is simple. Then the Malkin bifurcation function $M(\theta )$ associated to system (6) coincide with the Malkin bifurcation function ${M}_{\xi}(\theta )$ 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 ${C}_{0}$semigroups; see, e.g., [42].
where $(x,y)\in Q:=\mathrm{\Omega}\times (0,1)\subset {\mathbb{R}}^{n+1}$, $n\ge 2$, Ω is a ${C}^{2}$smooth bounded domain in ${\mathbb{R}}^{n}$, 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, ${\alpha}_{1}$, ${\alpha}_{2}$, ${\beta}_{1}$, ${\beta}_{2}$ are positive constants and the functions ${\varphi}_{1}$, ${\varphi}_{2}$ are of class ${C}^{1}$ jointly in their arguments. The linear part of system (26) generates an exponentially stable ${C}_{0}$semigroup in a suitable Banach space; see [43] as well as the related references therein. Under the assumption of the existence of a ${T}_{0}$periodic solution ${u}_{0}=({u}_{1}^{0},{u}_{2}^{0})$ of the limit problem, obtained as $\lambda \to 0$, and suitable conditions on the growth of the derivatives of ${\varphi}_{1}$, ${\varphi}_{2}$ with respect to their arguments, it is shown in [44] that [[43], Theorem 1] applies. This result guarantees the existence of ${\lambda}_{0}>0$ such that, for fixed $\stackrel{\u02c6}{\lambda}\in (0,{\lambda}_{0})$, system (26) has an isolated ${T}_{\stackrel{\u02c6}{\lambda}}$periodic solution ${u}_{\stackrel{\u02c6}{\lambda}}=({u}_{1}^{\stackrel{\u02c6}{\lambda}},{u}_{2}^{\stackrel{\u02c6}{\lambda}})$. The crucial assumption of [[43], Theorem 1] is that the zero eigenvalue of the linearized system around ${u}_{0}$ 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 [46].
If we assume that the superposition operators generated by the functions ${\varphi}_{1}$, ${\varphi}_{2}$, ${\psi}_{1}$, ${\psi}_{2}$ satisfy assumption (H2) of this paper, then (H1) and [[47], Theorem 4.3.1] ensure that the ${C}_{0}$semigroup generated by the linearization around ${u}_{\stackrel{\u02c6}{\lambda}}$ of the unperturbed system, corresponding to $\epsilon =0$ in (27), is strongly contractive with respect to the Hausdorff measure of noncompactness $\chi (\cdot )$, i.e., χstrongly contractive. Therefore, our abstract bifurcation result Theorem 2 applies to system (27).
Declarations
Acknowledgements
Dedicated to professor Jean Mawhin on the occasion of his seventieth birthday.
The first two authors acknowledge the support by RFBR Grants 100193112a and 12010392a. 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.
Authors’ Affiliations
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