An Invariant Set Bifurcation Theory for Nonautonomous Nonlinear Evolution Equations

In this paper we establish an invariant set bifurcation theory for the nonautonomous dynamical system $(\va_\lam,\0)_{X,\cH}$ generated by the evolution equation \be\label{e0}u_t+Au=\lam u+p(t,u),\hs p\in \cH=\cH[f(\.,u)]\ee on a Hilbert space $X$, where $A$ is a sectorial operator, $\lam$ is the bifurcation parameter, $f(\.,u):\R\ra X$ is translation compact, $f(t,0)\equiv0$ and $\cH[f]$ is the hull of $f(\.,u)$. Denote by $\va_\lam:=\va_\lam(t,p)u$ the cocycle semiflow generated by the equation. Under some other assumptions on $f$, we show that as the parameter $\lam$ crosses an eigenvalue $\lam_0\in\R$ of $A$, the system bifurcates from $0$ to a nonautonomous invariant set $B_\lam(\.)$ on one-sided neighborhood of $\lam_0$. Moreover, $$\lim_{\lam\ra\lam_0}H_{X^\a}\(B_\lam(p),0\)=0,\hs p\in P,$$ where $H_{X^\a}(\.,\.)$ denotes the Hausdorff semidistance in $X^\a$ (here $X^\alpha$ ($\a\geq0$) defined below is the fractional power spaces associated with $A$). Our result is based on the pullback attractor bifurcation on the local central invariant manifolds $\cM^\lam_{loc}(\.)$.


Introduction
Invariant set bifurcation theory of autonomous dynamical systems has been extremely well developed [1,6,12,13,15,[19][20][21][22][23][25][26][27]. A relatively simpler but important case is that of bifurcations from equilibria, including bifurcation to multiple equilibria (static bifurcation) and to periodic solutions (Hopf bifurcation) (see among others, [6,19]). Ma and Wang [20] and Sanjurjio [26] developed a local attractor bifurcation theory. Roughly speaking, if the trivial equilibrium e of an autonomous system changes from an attractor to a repeller on the local center manifold of the equilibrium when the bifurcation parameter λ crosses a critical value λ 0 , then the system bifurcates a compact invariant set K which is an attractor of the system restricted to the center manifold. Chow and Hale [6] started to discuss stability and bifurcation phenomena associated with more general invariant sets, e.g. periodic orbits. Using Conley index theory, Rybakowski [25] and Li and Wang [15] developed global bifurcation theorems to discuss bifurcation phenomena of nonlinear autonomous evolution equations. However, except some relatively simple nonautonomous cases, there are few papers studying the invariant set bifurcation for nonautonomous dynamical system. In [14] Langa et al. presented a collection of examples to illustrate bifurcation phenomena in nonautonomous ordinary differential equations. In [5] Carvalho et al. studied the structure of the pullback attractor for a nonautonomous version of the Chafee-Infante equation, and investigated the bifurcations that this attractor undergoes as bifurcation parameter varies.
Unlike autonomous dynamical systems for which forward dynamics is studied, pullback dynamics is much more natural than the more familiar forward dynamics for nonautonomous dynamical systems. This makes it very difficult to extend the invariant set bifurcation theory of autonomous systems to nonautonomous systems. Our approach in the paper is to treat the nonautonomous system as a cocycle semiflow over a suitable base space rather than a process. The biggest advantage of the cocycle semiflow framework is that in many cases the base spaces are compact, while the default base space R (real number set) for processes is unbounded. Based on the compactness of the base spaces, we can establish the equivalence between pullback attraction of cocycle semiflow and forward attraction of the associated autonomous semiflow. This device makes the dynamics of such a nonautonomous system appear like those of an autonomous system.
Without the compactness assumption on the base spaces, the upper semicontinuity of global pullback attractors for nonautonomous systems was obtained in Caraballo and Langa [2]. However, compact forward invariant sets of the perturbed systems are required to guarantee the existence of perturbed pullback attractors. In the paper, we suppose that the base spaces of cocycle semiflows considered are compact, which will require some restrictions on the nonlinearities. As a result, after introducing the notion of (local) pullback attractors (see Definition 2.5), we can establish a general result on the stability of local pullback attractors as the perturbation parameter is varied. Based on this result, a local pullback attractor bifurcation theory can be developed. This can be regarded as a nonautonomous generalization of autonomous attractor bifurcation theory in [20]. Finally, we study the bifurcation of invariant sets for the cocycle semiflow ϕ λ generated by the nonautonomous nonlinear evolution equation (0.1). We first construct a local central invariant manifold M λ loc (·) for ϕ λ with λ near λ 0 . Under further assumptions on f to ensure that 0 is a pullback attractor for ϕ λ 0 , we then restrict ϕ λ to M λ loc (·) and obtain a pullback attractor bifurcation on M λ loc (·) as λ crosses λ 0 . It leads to an invariant set bifurcation for ϕ λ . It is worth mentioning that if 0 is not an attractor but a repeller for ϕ λ 0 , our result still holds. Denote by B λ (·) the bifurcated invariant set. We further know that This paper is organized as follows. In Section 2, we present respectively some basic facts in autonomous and nonautonomous dynamical systems which will be required in the rest of the work. Section 3 deals with the stability of pullback attractors as bifurcation parameter varies. In Section 4, we establish an invariant set bifurcation theory for (0.1). We illustrate the main results with an example in Section 5. Finally, Section 6 contains the proofs of two propositions presented earlier in the paper.

Preliminaries
In this section we introduce some basic definitions and notions [7,8].
Let X be a complete metric space with metric d(·, ·). Given M ⊂ X, we denote M and int M the closure and interior of any subset M of X, respectively.
The Hausdorff semidistance in X is defined as

Semiflows and attractors
Let R + = [0, ∞). A continuous mapping S : R + × X → X is called a semiflow on X, if it satisfies i) S(0, x) = x for all x ∈ X; and ii) S(t + s, x) = S(t, S(s, x)) for all x ∈ X and t, s ∈ R + .
Let S be a given semiflow on X. As usual, we will rewrite S(t, x) as S(t)x.
Let B and C be subsets of X. We say that B attracts C under S, if Definition 2.1 A compact subset A ⊂ X is called an attractor for S, if it is invariant under S and attracts one of neighborhood of itself.
It is well known that if U is a compact positively invariant set of S, then the omega-limit set ω(U) := T ≥0 t≥T S(t)U is an attractor of S. The definition of the attraction basin of the attractor and other properties of local attractors can be found in [9,18,25].

Cocycle semiflows and pullback attractors
A nonautonomous system consists of a "base flow" and a "cocycle semiflow" that is in some sense driven by the base flow. A base flow {θ t } t∈R := {θ(t)} t∈R is a flow on a metric space P such that θ t P = P for all t ∈ R. Definition 2.2 A cocycle semiflow ϕ on the phase space X over θ is a continuous mapping ϕ : • ϕ(t + s, p, x) = ϕ(t, θ s p, ϕ(s, p, x)) (cocycle property).

Remark 2.3
If we replace R + by R in the above definition, then ϕ is called a cocycle flow on X.
We usually denote ϕ(t, p)x := ϕ(t, p, x). Then {ϕ(t, p)} t≥0, p∈P can be viewed as a family of continuous mappings on X.
For convenience in statement, a family of subsets B(·) := {B p } p∈P of X is called a nonautonomous set in X. As usual, we will rewrite B p as B(p), called the p-section of B(·). We also denote B the union of the sets B(p)×{p} (p ∈ P ), i.e., Note that B is a subset of X × P . A nonautonomous set B(·) is said to be closed (resp. open, compact), if each section B(p) is closed (resp. open, compact) in X. A nonautonomous set U(·) is called a neighborhood of B(·), if B(p) ⊂ int U(p) for each p ∈ P .
A nonautonomous set B(·) is said to be invariant (resp. forward invariant) under ϕ if for t ≥ 0, Let B(·) and C(·) be two nonautonomous subsets of X. We say that B(·) pullback attracts C(·) under ϕ if for any p ∈ P , Let ϕ be a given cocycle semiflow on X with driving system θ on base space P . The (autonomous) semiflow Φ := {Φ(t)} t≥0 on Y := P × X, given by is called the skew product semiflow associated to ϕ. The following fundamental result studies the relationship between the pullback attraction of ϕ and attraction of Φ. The proof is given in Appendixes.
Proposition 2.4 Let (ϕ, θ) X,P be a nonautonomous system, and let Φ be the skew-product flow associated to ϕ. Let K(·) and B(·) be two nonautonomous sets. Suppose P and K P := p∈P K(p) ⊂ X are both compact. Then K(·) pullback attracts B(·) through ϕ if and only if K := p∈P K(p) × {p} attracts B := p∈P B(p) × {p} through Φ.
Definition 2.5 Let (ϕ, θ) X,P be a nonautonomous system. A nonautonomous set A(·) is called a (local) pullback attractor for ϕ if it is compact, invariant and pullback attracts a neighborhood U(·) of itself.
The local pullback attractor defined here, very similar to the notion of a past attractor in Rasmussen [24], can be seen as a nature nonautonomous generalization of the local attractor from the autonomous theory. Similar to the case of autonomous systems, if U(·) is a compact forward invariant set of ϕ, then the omega-limit set ω(U)(·) defined as is a pullback attractor of ϕ. For instance, consider the following simple system on X = R: x where h(t) = 2 + sin t and H[h] is its hull which is the closure for the uniform convergence topology of the set of t-translates of h. The translation map θ t : H → H given by θ t p(s) = p(t + s) defines a flow on H. Then the unique solution of (2.1) define a cocycle flow on X given by provided that |x| ≤ 1/2. Therefore [−1/2, 1/2] is a forward invariant set of ϕ and it is pullback attracted by the pullback attractor 0. It is worth noting that 0 is only a local pullback attractor. Indeed, provided that |x| ≥ 2. It follows that 0 is only a local pullback attractor of ϕ.
In general, it is difficult to define the attraction basin of a pullback attractor. Fortunately, under the assumptions of Proposition 2.4, we can define the pullback attraction basin of a pullback attractor A(·). Specifically, we have Definition 2.6 Let (ϕ, θ) X,P be a nonautonomous system, and let Φ be the skew-product flow associated to ϕ. Suppose P is compact. Let A(·) be a pullback attractor of ϕ such that A P : A attracts (x, p) through Φ} be the attractor basin of A under Φ. Then the pullback attraction basin B(A)(·) of A(·) can be defined as

Stability of pullback attractors
We now establish a result on the stability of pullback attractors under a small perturbation. In fact, we prove a continuity result with respect to the Hausdorff semidistance.
Let X be a Banach space with norm · , and let A be a sectorial operator on X. Pick a number a > 0 such that Re σ(A + aI) > 0.
Denote Λ = A + aI. For each α ≥ 0, define the fractional power space as X α = D(Λ α ), which is equipped with the norm · α defined by Note that the definition of X α is independent of the choice of the number a. If A has compact resolvent, the inclusion X α ′ ֒→ X α is compact for α ′ > α ≥ 0.
(H2): For every T > 0 and compact subset B of X, we have Under the assumptions (H1), (H2), we can get a result on the stability of pullback attractors.
Assume U is a compact neighborhood of A λ 0 in Y = X × P , then under the assumptions (H1), (H2), the following statements hold.
( ) There exists a small δ > 0 such that for each λ ∈ I λ 0 (δ), ϕ λ has a pullback attractor A λ (·) such that Proof. ( ) By the compactness of U, we know that A λ 0 P := p∈P A λ 0 (p) is compact. Since A λ 0 (·) pullback attracts U(·) and P is compact, by Proposition 2.4, A λ 0 attracts U through Φ λ 0 . Since U is a neighborhood of A λ 0 , one knows that A λ 0 is an attractor of Φ λ 0 . By the assumption (H2), for any compact set B ⊂ X, we have that uniformly with respect to t ∈ [0, T ] and (x, p) ∈ B ×P . Then by the stability of the autonomous attractors [17, Theorem 4.1], there exists a δ > 0 (independent of p ∈ P ) such that for each λ ∈ I λ 0 (δ) : To complete the proof of ( ), we shall prove (6.16) by contradiction. Thus, let us assume that there exist σ > 0 and sequences λ j → λ 0 , as j → ∞, Note that x j = ϕ λ j (n, θ −n p)x n j , for some x n j ∈ A λ j (θ −n p). Similar to the argument in ( ), we can assume that A λ j (p) ⊂ U(p), thus x j ∈ U(p). By the compactness of U(p), there exists a subsequence of x j (still denoted by x j ) which converges to some x 0 ∈ U(p). Now, for each fixed n we have x n j ∈ U(θ −n p) so that there is a further subsequence of x n j (still denoted by x n j ) which converges to some x n 0 ∈ U(θ −n p). On the other hand, for any given ν > 0, we can use the assumption (H2) and the continuity of ϕ(n, θ −n p) to show that for j large enough, Then, for each fixed n ∈ N, Since U(p) is forward invariant, we have which contradicts (3.6). The proof is complete.
The main contribution of Theorem 3.1 is the existence of pullback attractor A λ (·) for ϕ λ as λ near λ 0 , while the argument of the upper semicontinuity of pullback attractors is an adaptation of that of [2].
The conditions of the following results may be easier to be verified in applications.

Invariant set bifurcation for nonautonomous nonlinear evolution equations
Based on the general result of the stability of pullback attractors, in the section we can establish some results on invariant set bifurcation for nonautonomous dynamical systems.

Problem and mathematical setting
From now on, we assume X is a Hilbert space with inner product (·, ·). We will consider and study invariant set bifurcation of the evolution equation on X, where λ ∈ R is a bifurcation parameter, the nonlinearity f : R×X α → X is bounded continuous mapping satisfying uniformly on t ∈ R. Moreover, there is β > 0 such that for t ∈ R and u ∈ X α , where κ : X → R + is a nonnegative function satisfying that κ(u) = 0 if and only if u = 0.
Denote k(ρ) the Lipschitz constant of f (t, ·) in B X α (ρ). Then by (4.2), Denote C b (R, X) the set of bounded continuous functions from R to X. Equip C b (R, X) with either the uniform convergence topology generated by the metric or the compact-open topology generated by the metric .
In application, f (·, u) is often taken as a periodic function, quasiperiodic function, almost periodic function, local almost periodic function [7,16] or uniformly almost automorphic function [28]. In this case, the hull H is a compact metric space. Accordingly, the translation group θ on H is given by θ t p(·, u) = p(t + ·, u), t ∈ R, p ∈ H.
Instead of (3.2), we will consider the more general cocycle system in X α (where α ∈ [0, 1)): u t + Au = λu + p(t, u), p ∈ H. (4.5) Proposition 4.1 [10] Let A and p be given as above. Assume that p is locally Hölder continuous in t. Then for each u 0 ∈ X α , there is a T > t 0 such that (4.5) has a unique solution For convenience, from now on we always assume that the unique solution (4.6) is globally defined. Define ϕ λ (t, p)u := u λ (t, 0; u, p), u ∈ X α , p ∈ H.
Then ϕ λ is a cocycle semiflow on X α driven by the base flow θ on H. Note that for each p ∈ H, u(t) is a p-solution of ϕ λ on an interval J if and only if it solves the equation (4.5) on J.

Invariant set bifurcation
Firstly, let us restrict the equation (4.5) on the invariant manifold M λ loc (·), λ ∈ I λ 0 (η/8). Specifically, we study the finite dimensional equation Denote φ λ the cocycle flow on B X α c (̺) with driving system θ on the base space H generated by (4.9).
Proof. Since X α c is finite dimensional, all the norms on X α c are equivalent. Hence for convenience, we equip X α c the norm · of X in the following argument.
Henceforth we will suppose that (F3) The hull H is a compact metric space.
Let λ ∈ I + λ 0 (η ′ ) be fixed, and let w(t) = y(−t). Then w(t) satisfies Taking the inner product of the equation (4.16) with w in X α , we have where c > 0 only depends on α. We get from (4.17) and (4.18) that for sufficiently small w α , which shows for fixed λ ∈ I + λ 0 (η ′ ), 0 locally asymptotically stable for the cocycle flow generated by the equation (4.16). In other words, 0 is a repeller of φ λ when λ ∈ I + λ 0 (η ′ ) and repels a neighborhood of 0 in X α c . This implies that 0 / ∈ A λ (·), λ ∈ I + λ 0 (η ′ ). The proof is complete. We are now in position to give and prove the main result of this paper concerning the invariant set bifurcation of ϕ λ . Proof. Let A λ (·) be the bifurcated attractor obtained in Theorem 4.5. Define We know from Theorem 4.5 that 0 / ∈ B λ (·) and B λ (·) ⊂ M λ loc (·). Based on the compactness of A λ (p) and the continuity of ξ λ p (y) in y, we can directly derive the compactness of B λ (p). So B λ (·) is compact.

An example
Consider the nonautonomous system: where Ω is a bounded domain in R 3 with smooth boundary, h is a function such that h(t) ≥ δ > 0 for some δ > 0. Denote by A the operator −∆ associated with the homogeneous Dirichlet boundary condition. Then A is a sectorial operator on X = L 2 (Ω) with compact resolvent, and D(A) = H 2 (Ω) H 1 0 (Ω). Note that A has eigenvalues 0 < µ 1 < µ 2 < · · · < µ k < · · · . Denote V = H 1 0 (Ω). By (·, ·) and | · | we denote the usual inner product and norm on H, respectively. The inner product and norm on V , denoted by ((·, ·)) and · , respectively, are defined as The system (5.1) can be written into an abstract equation on X: Define the hull H := H[h(·)u 3 ]. By the assumption on h, it is clear that Consider the cocycle system: Denote ϕ ± λ := ϕ ± λ (t, p)u the cocycle semiflow on H 1 0 (Ω) driven by the base flow (translation group) θ on H.
Since all the hypotheses in the main theorem above are fulfilled, we obtain some interesting results concerning the dynamics of the perturbed system. In particular, Theorem 5.1 Suppose H is compact. Then the cocycle semiflow ϕ − λ (resp. ϕ + λ ) bifurcates from (0, µ k ), k = 1, 2, · · · an invariant compact set B − λ (·) for λ > λ 0 (resp. B + λ (·) for λ < λ 0 ) and for each p ∈ P , lim resp. lim This means the compact set P × K P attracts B through Φ. Therefore the omega-limit set ω(B) of B exists and attracts B.
In the following, we prove ω(B) ⊂ K, which completes the necessity. For this purpose, define a nonautonomous setB(·) as follows It is clear that B(·) ⊂B(·). We first sayB(·) is forward invariant. Indeed, for any t ≥ 0 and p ∈ P , SoB(·) is forward invariant, which implies the omega-limit set ω(B)(·) ofB(·) is the maximal invariant set inB(·). Furthermore, for each p ∈ P , where the third "=" holds since for each fixed t ≥ 0 and p ∈ P , ϕ(t, θ −t p) is a continuous map on X. It follows that ω(B)(·) is the maximal forward invariant set inB(·). Therefore C : Finally, by the assumption that K(·) attracts B(·), one knows that ω(B)(·) ⊂ K(·), and thus C ⊂ K, which shows ω(B) ⊂ K.

Sufficiency:
In a very similar way as above, we can prove the sufficiency. By the compactness of P , which implies ω(B)(·) exists and pullback attracts B(·).

Construction of local invariant manifold
Let M > 0. For µ ≥ 0, define a Banach space as which is equipped with the norm · Xµ , According to the spectral decomposition, the space X has a direct sum decomposition: X = X − ⊕ X c ⊕ X + . Denote X ± := X − X + . Note that each be the projection from X to X i . Denote A λ i = A λ | X i . By the assumption (F2), we deduce that if λ ∈ (λ 0 − η/4, λ 0 + η/4) then for α ∈ [0, 1), Proof of Proposition 4.2. Let χ : R → R be a smooth function such that For ρ > 0, one can then define a smooth function such that Select suitable χ such that where k(ρ) is the local Lipschitz constant of f given in (4.4). Instead of (4.5), we consider the truncated system Suppose that ρ is so small that Let u ∈ X η/2 . By simple computations, we know that u is the solution of (6.7) if and only if it solves the integral equation Let p ∈ H and λ ∈ I λ 0 (η/8) be fixed. For each y ∈ B X α c (̺), one can use the righthand side of equation (6.9) to define a contraction mapping T := T y on X η/2 as follows: We first verify that T maps X η/2 into itself. For notational convenience, we write 0 ∧ t = min{0, t}, 0 ∨ t = max{0, t}, for t ∈ R.
Thanks to the Banach fixed-point theorem, T has a unique fixed point γ y p,λ ∈ X η/2 which is precisely a full solution of (4.5) with Π c γ y p,λ (0) = y and solves the integral equation γ y p,λ (t) = e −A λ c t y + t 0 e −A λ c (t−τ ) Π c p ρ (τ, γ y p,λ (τ ))dτ where the differential mean value is applied to e (λ 2 −λ 1 )t −1 to get the last term. It is clear that the integral and thus ξ · p (y) is Lipschitz continuous on I λ 0 (η/8) uniformly on p ∈ P and y ∈ B X α c (̺).