ON A DEFINITION OF MORSE-SMALE EVOLUTION PROCESSES

In this paper we consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework. In particular we consider nonautonomous perturbations of autonomous systems. In this case our definition of Morse-Smale evolution process holds for perturbations of Morse-Smale autonomous systems with or without periodic orbits. We establish that small nonautonomous perturbations of autonomous Morse-Smale evolution processes derived from certain nonautonomous differential equations are Morse-Smale evolution processes. We apply our results to examples of scalar parabolic semilinear differential equations generating evolution processes and possessing periodic orbits.


1.
Introduction. In recent years there is an increasing interest in the study of nonautonomous evolution equations expressed either by ordinary or partial differential equations. In the mathematical literature the main object of these studies is the long time behavior of the solutions of these systems, see [11,2,30,8]. The appropriate notions of attractor in the nonautonomous framework have been considered, and the study of their continuity and structural stability properties is still going on. Here the study of nonautonomous perturbations of autonomous systems is of particular relevance. Bortolan, Carvalho and Langa addressed these questions in [4]. They introduce a notion of Morse-Smale evolution process which is suitable for dynamically gradient evolution processes. This approach, reminiscent of Conley's treatment of isolated invariant sets, [15], is quite appropriate for gradient-like flows. However, if the unperturbed autonomous system contains periodic orbits this approach is not adequate unless we consider the collapse of the corresponding invariant sets. Since the classical notion of Morse-Smale dynamical system (in the autonomous framework) also involves the case of (finitely many hyperbolic) periodic orbits, it is appropriate to consider nonautonomous perturbations of autonomous systems possessing periodic orbits. In this paper we consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework.
A Morse-Smale dynamical system φ 0 : R × B → B has a finite number of hyperbolic equilibria and also a finite number of hyperbolic periodic orbits. Let S 0 (·, ·) denote the corresponding autonomous evolution process in B and consider the nonautonomous evolution processes S ε (·, ·) corresponding to its (nonautonomous) perturbations. Under a small ε-perturbation the solutions corresponding to the hyperbolic equilibria of φ 0 become hyperbolic solutions of the nonautonomous evolution process S ε (·, ·), [5]. But to understand the nonautonomous perturbations of a hyperbolic periodic orbit we need to consider the corresponding (normally hyperbolic) isolated invariant manifolds. This is illustrated in Section 2 by a simple example of an ordinary differential equation on B = R 2 . We consider nonautonomous perturbations of an autonomous ODE with one hyperbolic periodic orbit and describe the perturbations of the isolated invariant cylinder in R × B corresponding to the periodic solutions of the unperturbed system. In particular, we illustrate the existence of two special types of perturbations (normal and tangential) for these invariant cylinders.
In Section 3 we recall the notions of semiflows, global orbits, invariant sets and global attractors. We also recall the extension of these notions to the nonautonomous setting and we use the induced semiflows on R × B to compare the evolution processes.
The central notion of pullback attractor and its continuity properties is considered in Section 5.
In Section 6 we recall the notions of exponential dichotomy and trichotomy associated to hyperbolic and partially hyperbolic solutions of evolution process. We consider evolution process admitting a pullback attractor and satisfying well-known reversibility properties. We also recall the notions of stable and unstable manifolds and consider the normally hyperbolic isolated invariant manifolds associated to (perturbations of) hyperbolic periodic orbits.
In Section 7 we recall and collect the classic results on the behavior of hyperbolic and normally hyperbolic isolated invariant manifolds under perturbation. The study of conditions which ensure that such invariant manifolds are preserved under perturbations are abundant in the literature. See for example Sacker [36], Fenichel [18] and Hirsch, Pugh and Shub [27]. Quoting Pliss and Sell, [35]: "The fact that a normally hyperbolic invariant manifold persists under a small C 1 -perturbation is, of course, well known". Here we collect some results on the behavior of certain isolated invariant manifolds for differential equations under nonautonomous perturbations. We essentially recall the classic results on integral manifolds for perturbed differential systems (the Krylov-Bogoliubov method) following Bogoliubov and Mitropolski [3], Hale [20,21,22], Coppel and Palmer [16], and Henry [25].
The definition of Morse-Smale systems on B involves the concept of nonwandering behavior for hyperbolic critical orbits. For the case of processes, we instead introduce in Section 8 the notion of recurrent behavior on R × B neighborhoods of graphs of global solutions z(·) (simply referred to as recurrent trajectories). We also introduce the notion of recurrent behavior outside a manifold M z . This notion is motivated by the behavior of normally hyperbolic invariant manifolds corresponding to periodic orbits under normal/tangential perturbations as considered in Section 2.
Using the notion of recurrent behavior we define Morse-Smale evolution process in Section 9. Then we show in Theorem 9.2 that autonomous evolution processes corresponding to Morse-Smale semiflows on B are Morse-Smale evolution processes. Subsequently we consider evolution processes obtained by nonautonomous perturbations of Morse-Smale autonomous evolution processes. We specifically study processes derived from certain nonautonomous differential equations already analyzed in Section 7 for the persistence of isolated invariant manifolds. For these problems we establish in Theorem 9.3 the following openness property: small nonautonomous perturbations of autonomous Morse-Smale evolution processes are Morse-Smale evolution processes.
In the last Section we apply these results to examples of nonautonomous perturbations of autonomous Morse-Smale systems. In particular we consider scalar parabolic semilinear differential equations generating evolution processes on adequate fractional power spaces and possessing periodic orbits. We also consider the case of asymptotically autonomous evolution processes with a Morse-Smale limiting behavior.
The relevance of Morse-Smale systems for the discussion of structural stability of autonomous systems is well established in the literature. For nonautonomous systems, structural stability has also been considered; see for example [31] in the finite dimensional case, and [2] for local aspects in infinite dimensions.
The notion of Morse-Smale evolution process for a gradient like system introduced in [4] proved useful for the discussion of qualitative aspects of the dynamics related to structural stability of nonautonomous systems (see [9]). We believe that the notion of Morse-Smale evolution process introduced here is appropriate for the discussion of structural stability, but this topic should be addressed elsewhere.
2. Nonautonomous perturbations of autonomous ODEs. In this Section we describe examples of nonautonomous perturbations of an autonomous ODE with one hyperbolic periodic orbit. We consider the perturbations of the isolated invariant cylinder corresponding to the periodic solutions of the unperturbed system. We show that, while certain perturbations deform the cylinder into a tube fibered by nonhyperbolic trajectories (a normally hyperbolic invariant manifold, see Sections 6-7), other perturbations preserve the cylinder but perturb the flow on it producing hyperbolic behavior.
Let us start with the following example of a nonautonomous system of ordinary differential equations on B = R 2 , where δ is a real parameter and f 1 , f 2 are sufficiently smooth bounded functions. In polar coordinates, x = ρ cos θ, y = ρ sin θ, system (1) is given as with (f 1 , f 2 ) also in polar coordinates. For δ = 0 we obtain the unperturbed autonomous system This system possesses only one fixed point (0, 0) and one stable limit cycle Moreover, from the linearization of (3) around periodic solutions, it is known that µ = 1 is a simple characteristic multiplier and Π is a hyperbolic periodic orbit. In R × B the set of trajectories of periodic solutions of (3) forms a cylinder with Π as its natural projection on B.
For small δ = 0 we first consider a nonautonomous perturbation of the form where g : R → R is a smooth bounded function and ϕ : R + → [0, 1] is a smooth cutoff satisfying and monotone for r ∈ (1/4, 1/2) and r ∈ (2, 4). In polar coordinates this is which leads to the perturbed system This is a simpler case of the example in [4,Section 3.4] where it is pointed out that this perturbation deforms the cylinder into a tube containing an infinite number of trajectories of nonhyperbolic global solutions. In view of this deformation we say that a perturbation of the form (4) is a normal perturbation with respect to the invariant cylinder. Our next example uses a perturbation of the form which in polar coordinates is leading to the perturbed system As a consequence, the perturbation preserves the invariant cylinder corresponding to the periodic orbit, x, y) ∈ R × B : x 2 + y 2 = 1} , but changes completely the dynamics on the cylinder. In fact, for δ = 0 the perturbed flow on the cylinder contains exactly two trajectories of hyperbolic solutions of (1). To see this, we change our polar coordinates to a co-rotating frame with respect to the periodic orbit, (ρ, θ) → (ρ, ψ) with ψ = θ − t. This substitution changes only the second equation of (9) and we obtain the autonomous system in the new polar coordinates An easy integration shows that the solutions have the form We observe that if ρ 0 = 1, then ρ(t) = 1 for all t ∈ R and (with the usual (mod 2π) polar identifications) we can distinguish three equilibria of (10), namely (0, 0), (1, 0), and (1, π). In the coordinates (ρ, θ) these equilibria correspond to the solutions ξ 0 (t) = (0, t), ξ 1 (t) = (1, t), and ξ 2 (t) = (1, π + t) of (9). In the original coordinates (x, y) they correspond to the solutions ζ 0 (t) = (0, 0), ζ 1 (t) = (cos t, sin t), and ζ 2 (t) = (− cos t, − sin t) of (1).
Each bounded global orbit of the planar autonomous system (10) is either an equilibrium or a connecting orbit between two different equilibria. Since the equilibria have different Morse indices the solutions of (1) corresponding to the connecting orbits cannot be hyperbolic. Therefore, the nonautonomous system (1) admits only three hyperbolic solutions, ζ 0 , ζ 1 and ζ 2 .
In view of these observations we say that a perturbation of the form (7) is a tangential perturbation with respect to the invariant cylinder.
These examples illustrate two types of perturbations of invariant cylinders corresponding to periodic orbits: the normal perturbations which deform the cylinders into tubes fibered by an infinite number of trajectories of nonhyperbolic solutions, and the tangential perturbations which reduce the cylinders corresponding to periodic orbits to a finite set of trajectories of hyperbolic solutions and their heteroclinics. Therefore a definition of Morse-Smale nonautonomous system that takes into consideration the periodic orbits of autonomous systems has to comprise both types of perturbations.
3. Semiflows and evolution processes. We start by recalling the well-known notion of a semiflow.
One assumes also that the map For a semiflow one can define the notions of global orbits, invariant sets and global attractors, see e.g. [23], [38], [13]. The qualitative theory of dynamical systems involves the study of these global attractors and one of the essential tools available is the Morse-Smale property (see [34], [24] and [32]). Most of these notions introduced in the case of semiflows have been recently extended to the case of nonautonomous systems with the consideration of evolution processes and also skew product flows, see in particular [2], [30], [8] and the references therein. Here we will stay in the framework of evolution processes.
Let ∆ be the diagonal of R × R and consider ∆ + = {(t, τ ) ∈ R × R : t ≥ τ } which is a manifold with boundary ∆ and, moreover, is a groupoid acting on R. Furthermore, let F be a given metric space of parameters, which in applications will be suitably chosen depending on the problem in view.
Note that a nonautonomous differential equation on B, satisfying conditions of existence, uniqueness and global continuation of solutions, defines an evolution process on B.
called the global evolution process, is assumed to be continuous and it is used to analyze the dependence on f ∈ F for a given family of evolution processes.
In this case the evolution process S f is called an autonomous evolution process. Otherwise, S f is said to be nonau- it can be seen that S f is an autonomous evolution process on B (see [5], [10], [6]).
Definition 3.6. Under appropriate smoothness conditions of the global evolution process S(f, (t, τ ), x) we define its derivative with respect to t on the diagonal ∆ For instance, if we consider evolution processes generated by ODEs on B = R n , then D ∆ (S) = R × R n . For evolution processes generated by delay differential 4. Comparing evolution processes. In order to compare evolution processes, in particular autonomous with nonautonomous processes and if, moreover, we want to analyze some stable structures that appear in nonautonomous evolution processes on B, we will see in the sequel that any C k evolution process on B induces a C k semiflow on R × B.
For instance, the equation (1) for δ = 0 defines a Morse-Smale semiflow on B = R 2 . For δ > 0, the perturbed equation defines a nonautonomous process in the same space. Both processes can be considered and compared as semiflows on The main objective of the present paper is the introduction of a notion of Morse-Smale process which allows the comparison as semiflows of perturbed with unperturbed processes.
Proof. In fact, for any s ≥ 0 and (τ, and it is easy to see that T f is a C k semiflow (see [30,Theorem 2.4] and [2, p. 56]).
x] t=τ be defined as in Definition 3.6. Then, assuming that the domain D ∆ (S) has a differentiable structure, the semiflow above induces a vector field X f defined in R × B by Thus, the vector field X f defines the autonomous ODE in E = R×B of equations By eliminating time, one arrives to the nonautonomous system in B given by 5. Pullback attractors. A central object of the study of any dynamical system is the asymptotic behavior of its solutions. This leads to the consideration of a global attractor which under many forms is a notion pervasive in the literature. Hereafter we will use the following notion of pullback attractor, see [30,8] and the references therein.
where dist(·, ·) represents the Hausdorff semidistance, and is the minimal family of closed sets pullback attracting all bounded sets of B.
For our purposes it is essential to consider the characterization of pullback attractors for evolution processes and their behavior under perturbation. These topics have been considered in the literature and here, for completion, we only restate the needed results. The next Proposition follows essentially from [ Proposition 2. If a pullback attractor {A f (t) ⊂ B, t ∈ R} does exist for an evolution process S f (·, ·) on B and t∈R A f (t) is bounded in B, then, for each t ∈ R, we have that A f (t) is the set of all values z(t) where z : R → B is a global and bounded solution for S f (·, ·). Moreover, if S f (·, ·) is autonomous, then each A f (t) of the family coincides with the global attractor A f of the associated semiflow.
To study the dependence of global or pullback attractors on the parameter f ∈ F the appropriate notion is upper semicontinuity. The following proposition is motivated by [30,Theorem 3.36]. See also [8, Proposition 1.20 and Theorem 3.6].
Proposition 3. Suppose that the global evolution process S : 6. Hyperbolic and partially hyperbolic solutions. Consider an evolution process S f (t, τ ) as in Definition 3.2 and let F f (τ, x) be defined as in (15).
is said to be hyperbolic if the linear equation  [14] and [5]), then z 0 has a positively invariant local stable C 1 manifold and a negatively invariant local unstable C 1 manifold, the invariance being under the semiflow T f on E = R × B, and both manifolds contain the graph α = {(τ, z 0 (τ )) : τ ∈ R} of z 0 ; one denotes them by W s loc (α) and W u loc (α), respectively. We assume in the following that any evolution process S f (·, ·) considered admits a pullback attractor satisfying the hypotheses of Corollary 1, Propositions 2 and 3. In addition, we suppose that S f (·, ·) has the following reversibility property: S f (t, τ ) restricted to A f (τ ) is injective onto A f (t) for all t ≥ τ ∈ R and all f ∈ F, (hence a homeomorphism); also the derivative of S f (t, τ ) restricted to A f (τ ) is injective. Definition 6.2. Applying the semiflow T f to W u loc (α) one obtains the (global) unstable manifold W u (α) of z 0 which is, in general, immersed in R × B.
If instead, the linear equation (17) corresponding to a global and bounded solution z 0 of (16) admits an exponential trichotomy (see [2, Section 8.1]), the solution z 0 is said to be partially hyperbolic. In this case, under adequate conditions on F f (τ, x), z 0 possesses a center manifold invariant under the semiflow T f , see [2,Chapter 8], [40,12]. Here we will consider isolated invariant manifolds that are normally hyperbolic in the sense that the linearization around each solution with trajectory in the manifold has an exponential trichotomy with a two dimensional center (or intermediate, [2, p. 172 Remark that all these manifolds in R × B have dimension at least 1 since they must contain the graph α = {(t, z 0 (t)) : t ∈ R} of the trajectory corresponding to the solution z 0 . Hence, hyperbolicity of the solution z 0 corresponds to a center manifold of dimension r = 1. A periodic orbit with minimum period T > 0 of a semiflow T f is hyperbolic if the corresponding Poincaré map has exactly one simple multiplier µ = 1. In this case, the solution z 0 of the autonomous evolution process S f (t, τ ) = T f (t − τ ), t ≥ τ , is partially hyperbolic. The hyperbolic periodic orbit of T f generates a cylinder in R × B composed of graphs of the solutions z τ (·) = T f (τ )z 0 (·), 0 ≤ τ < T , hence defining a center manifold of dimension r = 2 embedded in R × B which is normally hyperbolic. 7. Isolated invariant manifolds. In the following we collect some results on the behavior of certain isolated invariant manifolds for differential equations under nonautonomous perturbations. We consider specifically invariant manifolds associated to hyperbolic equilibria and hyperbolic periodic orbits. 7.1. Hyperbolic equilibria. Let Y ⊆ X be Banach spaces. Assume L is a linear operator generating a C 0 -semigroup of bounded linear operators in X and let g ∈ C 1 (Y, X). We consider either the finite dimensional case Y = X = R n or, in infinite dimensions, the case of −L sectorial and Y = X α , the fractional power space of −L with appropriate α ∈ (0, 1). We denote by C ε (R, C 1 (Y, X)) the metric space of continuous functions f = f (t, z) uniformly Lipschitz continuous for t ∈ R and differentiable in z with uniform ε-bounded C 1 -norm, i.e.
Let z 0 denote a hyperbolic equilibrium of the autonomous systeṁ Then for f ε ∈ C ε (R, C 1 (Y, X)) the nonautonomous perturbed equatioṅ has the following properties: There is an ε 0 > 0 such that for each 0 < ε ≤ ε 0 the nonautonomous system (19) has a global solution z ε = z ε (t) uniformly close to z 0 . This follows from the fixed point property of the variation of constants equation associated to (19) using the hyperbolicity of z 0 and the projections onto its stable and unstable manifolds.
Remark that the trajectory (t, z 0 ) corresponds to the hyperbolic solution of (19) ε=0 and the trajectory (t, z ε (t)) is the corresponding perturbed isolated invariant manifold.
According to Definition 3.2 and Remark 1 equation (19) defines an evolution process on B = Y with F = C ε0 (R, C 1 (Y, X)).
Then for f ε ∈ C ε (R, C 1 (Y, X)) the nonautonomous perturbed equation (19) satisfies the following: There is an ε 0 > 0 such that for each 0 < ε ≤ ε 0 the nonautonomous system (19) has an isolated invariant manifold M ε uniformly close to M 0 of the form Moreover M ε → M 0 as ε → 0 and is normally hyperbolic since the linearization of (19) around each solution ξ ε (·, τ ) with trajectory in M ε has an exponential trichotomy with a two dimensional center manifold.
The persistence of these isolated invariant manifolds is supported by the following results: For z ∈ R n , by [22,39] there is a local change of coordinates around ξ 0 (·) of the form z = ξ 0 (θ) + Q(θ)ρ , with (θ, ρ) ∈ R × R n−1 which applied to (19) yields

Hence system (20) is equivalent to
where, by the hyperbolicity of ξ 0 (·), there is a real α 0 > 0 such that the matrices B 1 and B 2 satisfy with G 1 , G 2 , continuous of class C 1 in θ and satisfying G 1 (t, θ, 0) = G 2 (t, θ, 0) = 0. Moreover, M ε has a saddle-point structure in the sense that any solution of (21) is such that r 1 → 0 exponentially as t → −∞ and r 2 → 0 exponentially as t → ∞ (both with asymptotic phase), see [22,Theorem VII.7.1]. This shows the existence of the isolated invariant manifold M ε of (19) with ξ ε (t, θ) = ξ 0 (θ) + Q(θ)F (t, θ, ε) , contained in a tubular neighborhood of the cylinder M 0 . The exponential trichotomy of each solution ξ ε (·, θ) follows from the saddle-point property of M ε . We also observe that the flow of (19) on M ε is equivalent to the equation For z ∈ X α , by [25,Section 9.2] there is a local change of coordinates around ξ 0 (·) generalizing the result of [22,39]. In fact, by [25,Theorem 9.2.2] there exists a closed subspace Y 0 ⊂ X of codimension 1 and a C 2 map Q : Γ → L(X) such that for each y ∈ Γ the linear map Q(y) is an isomorphism of Y 0 onto its image with Q(y)Y 0 ⊕ T y Γ = X, and a neighborhood of Γ has the form Then, assuming t → ξ 0 (t) ∈ X is C 2 and using y = ξ 0 (s), we introduce in N Γ the coordinates (s, x) and write for x < δ, whereQ(s) = Q(ξ 0 (s)) and s →Q(s) is of class C 2 and periodic with period ω > 0.
Using these coordinates, the flow of (18) in the neighborhood N Γ has the form ṡ = 1 + S(s, x) where the linear operator B with D(B) = D(L) ∩ Y 0 is sectorial in Y 0 , and B 0 (s) ∈ L(Y 0 ) is a family of bounded linear operators ω-periodic in s. Moreover, defining Y α 0 = Y 0 ∩ X α , we have that S and R are C 1 functions of s ∈ R and x ∈ Y α 0 , x α <δ, which are also ω-periodic in s and vanish for x = 0, see [25,Section 9.2]. In fact, by an appropriate choice of projection P ∈ L(X), with N (P ) ⊂ Y 0 , derived from an extension of Whitney's embedding theorem (see [25,Theorem 9.2.1]), we have B = (I − P )L|Y 0 , which implies that B is sectorial. Similarly, the flow of (19) in the neighborhood N Γ becomes ṡ = 1 + S(t, s, x) with C 1 functions S(t, s, x) and R(t, s, x) of the form Let T (t, τ ), t ≥ τ , denote the family of evolution operators on X generated by the linearization of (18) around ξ 0 (t), that is T (t, τ )z 0 = z(t, τ, z 0 ) for z 0 ∈ X where z(t) = z(t, τ, z 0 ), t ≥ τ , solveṡ z = Lz + g z (ξ 0 (t))z , z(τ ) = z 0 .
The characteristic multipliers of ξ 0 (t) are the nonzero eigenvalues of the Poincaré map U (t) = T (t + ω, t) .
We recall that the eigenvalues of the Poincaré map are independent of t and if L has compact resolvent then σ(U (·)) \ {0} is composed entirely of eigenvalues µ j , j = 1, 2, . . . , [25,Lemma 7.2.2]. Due to hyperbolicity, µ = 1 is a simple characteristic multiplier of ξ 0 (t), with the remaining spectrum of U (·) away from the unit circle. Moreover, the multipliers = 1 are exactly the multipliers of the linear equation dx ds = Bx + B 0 (s)x corresponding to the linearization around x = 0 of the second equation in (22). This equation has no multipliers on the unit circle and, therefore, has an exponential dichotomy. See [25, Section 9.3] for details.
Let T (·, ·) denote the linear evolution process corresponding to dx ds = Bx+B 0 (s)x and consider the Poincaré map U (s) = T (s + ω, s). By [25,Theorem 7.2.3], if σ 1 is a spectral set of σ( U (·)) \ {0} then, for each s the space Y 0 decomposes as Y 0 = X 1 (s) ⊕ X 2 (s), the direct sum of closed invariant subspaces under U (s), and where T (s 2 , s 1 ), s 2 ≥ s 1 , maps X 1 (s 1 ) one-to-one onto X 1 (s 2 ). Moreover, there is a family of bounded invertible linear operators P 1 (s) : X 1 (s 0 ) → X 1 (s), with period ω in s and P 1 (s 0 ) = Id, and a bounded linear operator C on X 1 (s 0 ) with e ωσ(C) = σ 1 such that T (s 2 , s 1 )x = P 1 (s 2 )e C(s2−s1) P −1 1 (s 1 )x , for all x ∈ X 1 (s 1 ) and all s 2 ≥ s 1 . Then, using a decomposition x = x 1 + x 2 ∈ X 1 (s) ⊕ X 2 (s) and letting x 1 = P 1 (s)y with y ∈ X 1 (s 0 ), the above system (23) is equivalent to       ṡ = 1 + S F (t, s, y, x 2 ) y = Cy +R F (t, s, y, x 2 ) where D(s) is a sectorial operator in X 2 (s), see [25,Section 7.2 ex.2]. This again brings about the role of the characteristic multipliers in the discussion of the asymptotic behavior of the flow in the neighborhood N Γ . Then, by a direct application of [25, Theorem 9.1.1] system (23) has an isolated invariant manifold of the form with continuous G of class C 1 in s and G(t, s, 0) = 0 as required. Moreover, the saddle-point nature of this manifold is exposed in (24) by choosing the spectral set σ 1 as the set of all multipliers outside the unit circle. It also exhibits the exponential trichotomy of each global solution ξ ε (·, s) of (19) with trajectory in M ε .
8. Recurrent behavior. The definition of Morse-Smale systems on B involves the concept of nonwandering behavior for hyperbolic critical orbits, see [34,24,32]. For the case of processes, we instead introduce the notion of recurrent behavior on R×B neighborhoods of graphs of global solutions (trajectories).
Let B(l, 0) be the ball in B of radius l > 0 centered at the origin and let ξ(t) = (t, z(t)) be the trajectory in R × B corresponding to a bounded global solution z in B. When the process has a pullback attractor, one defines the neighborhood U (t, l, z(t)) of z(t) in A f (t) by the expression U (t, l, z(t)) := {z(t) + B(l, 0)} ∩ A f (t) , t ∈ R . Definition 8.1. A trajectory ξ : R → R×B associated to a global bounded solution z(·) of (16) given by ξ(t) := (t, z(t)), t ∈ R, is said to produce a recurrent behavior of S f (·, ·) (or of T f (·)) if given a timet ∈ R and a neighborhood U (t, l, z(t)) of z(t) in A f (t) ⊂ B there existsz ∈ U (t, l, z(t)) withz = z(t) such that for any T > 0 the solutionz(·) of (16) withz(t) =z satisfiesz(t ± ) ∈ U (t ± , l, z(t ± )) for some t + >t + T and t − <t − T . For simplicity, in this case we say that the trajectory ξ : R → R × B is recurrent.
This definition of recurrent behavior is illustrated in Figure 2, whereξ := (t,z(t)) and U τ := U (τ, l, z(τ )). Figure 2. Recurrent behavior for ξ = (t, z(t)). Here t+ −t > T and Definition 8.2. Consider a global bounded solution z(·) of (16) and let M z denote a manifold containing the trajectory ξ(t) = (t, z(t)), t ∈ R, such that M z is invariant under the semiflow T f on R × B and has t-section M t z := M z ∩ ({t} × B) with natural B-projection P B M t z contained in A f (t). The trajectory ξ is said to produce a recurrent behavior of S f (·, ·) in the complement of M z if given a timet ∈ R and a neighborhood U (t, l, z(t)) of z(t) in A f (t) ⊂ B there existsz ∈ A f (t) \ P B Mt z (hencē z = z(t)) such that for any T > 0 the solutionz(·) of (16) withz(t) =z satisfies z(t ± ) ∈ U (t ± , l, z(t ± )) for some t + >t + T and t − <t − T . For simplicity, in this case we say that the trajectory ξ : R → R × B is recurrent outside M z .
Remark 3. A trajectory in R×R n of an autonomous evolution process corresponding to a constant hyperbolic solution in R n with a homoclinic orbit is recurrent. On the other hand, any solution of an autonomous evolution process corresponding to a constant solution of a gradient Morse-Smale flow in R n defines a nonrecurrent trajectory in R × R n .

Remark 4.
In contrast with the second observation of the previous remark, a trajectory in R × R n corresponding to a periodic solution in R n with a hyperbolic orbit is always recurrent. In fact, the process exhibits recurrent behavior on the invariant cylinder generated by the periodic orbit, which is the center manifold in R × R n of the partially hyperbolic periodic solution.
Example 1. A particular case of that special situation is given by the autonomous Morse-Smale ODE system on B = R 2 , considered in (3) of Section 2: The cylinder in this case is the subset of R × R 2 given by R × {(x, y) : x 2 + y 2 = 1} which is an invariant manifold for the lifted system: Observe that all the global solutions of (26) corresponding to the periodic solutions of (25) are partially hyperbolic and their graphs are fibers of the same normally hyperbolic cylinder, i.e., the union of all graphs of trajectories (t, z τ (t)) corresponding to the periodic solutions z τ (t) = (cos(t + τ ), sin(t + τ )) in R 2 for τ ∈ [0, 2π). Moreover, the recurrent behavior is restricted to this cylinder. In fact, any global solution (t, z(t)) of the lifted system (26) different from the trivial solution (t, 0, 0) and not in the cylinder is in the stable manifold of a solution corresponding to a periodic orbit, and z(t) either goes to (0, 0) or becomes unbounded as t → −∞. Therefore, if (t, z(t)) denotes a normally hyperbolic solution on the cylinder, the process generated by (25) when restricted to the complement of the cylinder has no recurrent behavior, although (t, z(t)) produces recurrent behavior on the cylinder. Moreover, this recurrent behavior on the cylinder is mostly preserved by the tangential components of generic perturbations, as illustrated in Section 2. This suggests that in the case of solutions on normally hyperbolic manifolds we should only study the (non)occurrence of recurrent behavior outside the center manifolds, since on the center manifolds recurrent behavior is always expected.
9. Morse-Smale evolution processes. According to Definition 3.3 the evolution process S f (·, ·) depends continuously on f ∈ F where the parameter space F is a metric space appropriately chosen depending on each problem. Recall that for each f ∈ F the evolution process S f (·, ·) is assumed to: (i) admit a pullback attractor satisfying the hypotheses of Propositions 2, 3; (ii) have the reversibility property, (see Section 6).
Our definition of Morse-Smale process follows from the previous observations regarding the occurrence and location of recurrent behavior in Example 1. We start by identifying the sets of global solutions which are relevant for this definition.
We let M h f ⊂ R×B denote the set of all graphs of hyperbolic solutions of S f (·, ·), M h f = {(t, z(t)) : z(·) hyperbolic solution of S f (·, ·), t ∈ R} . We also let M nh f denote the set of all normally hyperbolic isolated invariant manifolds M T f satisfying the following: (T1) M T f are fibered by graphs of partially hyperbolic solutions of S f (·, ·) possessing exponential trichotomy with two dimensional center manifold; (T2) for each t ∈ R the natural B-projection of the t-section of M T f (which is in A f (t)) is homeomorphic to a circle. Such invariant manifolds will be called tubes hereafter.
For each tube M T f we define its unstable manifold, W u (M T f ), as the union of all the unstable manifolds of its fibers α ∈ M T f , Similarly we define the local stable manifold, W s loc (M T f ). different critical elements c 1 , c 2 of T f . In fact, for c 1 = c 2 = c we cannot have W u (c) ∩ W s loc (c) c due to transversality since, in that case, the λ-lemma (see [33,32,24]) implies infiniteness of Ω(f ). Next, if c 1 and c 2 are distinct fixed points of T f , transversality between W u (c 1 ) and W s loc (c 2 ) also implies the strict inequality dim W u (c 1 ) > codim W s loc (c 2 ) since W u (c 1 ) ∩ W s loc (c 2 ) must contain the positive semi-orbit of γ in W s loc (c 2 ). Therefore we have This shows that the linearization of S f (·, ·) around z(·) cannot possess an exponential dichotomy which contradicts the hyperbolicity of z(·). By the same reason c 1 or c 2 cannot be hyperbolic periodic orbits of T f since the existence of a simple multiplier µ = 1 for the corresponding Poincaré map prevents hyperbolicity of z(·). Therefore we conclude that z(·) corresponds to a fixed point of T f . We now show that each tube in M nh f is a cylinder generated by a periodic orbit of T f . Let α = {(t, z(t)), t ∈ R} denote the graph of a partially hyperbolic solution z(·) of S f (·, ·) corresponding to a fiber of a tube M T f in M nh f . Repeating the previous arguments we have that z(·) is a solution of the semiflow T f with global bounded orbit γ in A f . Moreover the α-limit and ω-limit sets of the orbit γ are critical elements of the semiflow T f . We remark that since z(·) is partially hyperbolic γ cannot be a hyperbolic fixed point of T f . Also, the critical elements cannot be fixed points of T f ; in fact, if the α-limit or the ω-limit set of γ is a unit set of a fixed point, the linearization of S f (·, ·) around the solution z(·) cannot admit an exponential trichotomy with a two dimensional center manifold, since the fixed points are hyperbolic. This implies that the critical elements are periodic orbits, C 1 and C 2 . Next, arguing again by contradiction, we show that γ cannot connect two different periodic orbits. Assume that the periodic orbit C 1 is the α-limit set of γ, and C 2 is the ω-limit. Consider the Poincaré map for the semiflow T f defined at a local cross-section of C 1 , and let i(C 1 ) denote the number of multipliers µ j of this map outside the unit disk, |µ j | > 1, counting multiplicities. We recall that C 1 is hyperbolic and its unstable manifold is finite dimensional. Then we have dim W u loc (C 1 ) = i(C 1 ) + 1. Likewise, defining i(C 2 ) for the Poincaré map at a local cross-section of C 2 we have codim W s loc (C 2 ) = i(C 2 ) .
We now consider a point p ∈ γ on the intersection between the global unstable manifold of C 1 and the local stable manifold of C 2 , that is p ∈ W u (C 1 ) ∩ W s loc (C 2 ), and obviously dim W u (C 1 ) = i(C 1 ) + 1 . have C 1 as α-limit set and C 2 as ω-limit set. Hence the intersection W u (C 1 ) ∩ W s loc (C 2 ) contains positive semi-orbits of all the solutions corresponding to fibers of M T f . This implies that the transversal intersection W u (C 1 ) ∩ W s loc (C 2 ) is at least two dimensional and codim W s loc (C 2 ) ≤ dim W u (C 1 ) − 2 . From (28) and (29) we conclude that This shows that the linearization of S f (·, ·) around z(·) cannot possess an exponential trichotomy which contradicts the partial hyperbolicity of z(·). Therefore we must have C 1 = C 2 := C.
Again by transversality, invoking the λ-lemma and the finiteness of Ω(f ), we cannot have W u (C) ∩ W s loc (C) C. Therefore we conclude that the solution z(·) is a periodic solution of the autonomous semiflow T f and corresponds to the periodic orbit C. Hence the tube M T f is the cylinder generated by C, and M nh f is the set of all cylinders generated by periodic orbits of T f . This completes the proof of condition (1) of the Morse-Smale Definition 9.1. Condition (2) also follows because, for the autonomous evolution process, we have To prove condition (3) of the Morse-Smale Definition 9.1 we first consider the trajectory ξ(t) = (t, z 0 ), t ∈ R, corresponding to a hyperbolic fixed point z 0 of T f in B. For a given timet ∈ R let U (t, l, z 0 ) denote the neighborhood of z 0 in A f (t) ⊂ B with l > 0 sufficiently small (less than the minimal distance between z 0 and the other critical elements of T f ). Letz ∈ U (t, l, z 0 ) \ {z 0 }. Then the solutionz(·) of the autonomous evolution process withz(t) =z = z 0 is also a solution of the semiflow T f with the same initial conditionz(t) =z. Then, by the Morse-Smale property of the semiflow T f the α-limit set and ω-limit set of the orbit ofz(·) are two distinct critical elements of T f . Therefore at least one of these limit sets is at a distance from z 0 larger than l. This implies that the solutionz(·) leaves the neighborhood U (·, l, z 0 ) definitively either forward or backward in time, i.e., ξ(·) is nonrecurrent by Definition 8.1. Similar arguments show that trajectories corresponding to hyperbolic periodic orbits of T f are nonrecurrent outside their cylinders (Definition 8.2). This completes the proof. This Theorem establishes the class of Morse-Smale autonomous evolution processes on B. We now consider the evolution processes obtained by nonautonomous perturbations of the elements in this class. From here on we study processes derived from nonautonomous differential equations like the problems analyzed in Section 7 for the persistence of isolated invariant manifolds.
Let B ⊆ X be Banach spaces. Assume L is a linear operator generating a C 0semigroup of bounded linear operators in X and let g ∈ C 1 (B, X). As in Section 7, we consider either the finite dimensional case B = X = R n or, in infinite dimensions, the case of −L sectorial and B = X α , with appropriate α ∈ (0, 1). We also assume that the nonlinearity g ∈ C 1 (B, X) satisfies conditions ensuring the existence of pullback attractors {A 0 (t), t ∈ R}, {A ε (t), t ∈ R} for the evolution processes generated by the differential equations (18), (19), (equations (30), (31) below). Let F = C ε0 (R, C 1 (B, X)) be the metric space of parameters, for ε 0 sufficiently small. Theorem 9.3. If the autonomous evolution process on B generated bẏ is Morse-Smale, then for every f ε ∈ F the nonautonomous evolution process on B generated by the perturbed equatioṅ is also Morse-Smale. z = Lz + g(z) + f (t, z) , (33) with L and g satisfying the same conditions of Theorem 9.3, but with nonautonomous term f ∈ C b (R, C 1 (B, X)) satisfying lim t→+∞ sup z∈B f (t, z) X + f z (t, z) L(B,X) = 0 .
Then S(·, ·) is asymptotically autonomous and its limiting behavior should approach the autonomous evolution process S ω (·, ·) generated by the differential equatioṅ z = Lz + g(z) .
If we replace the time dependence in (33) by a smooth monotone function R t → h η (t) ∈ R satisfying we obtain an asymptotically autonomous evolution process which is autonomous in the past (for t ≤ η). Let S η (·, ·) denote the evolution process generated by the differential equationż = Lz + g(z) + f (h η (t), z) .