Lipschitz perturbations of Morse-Smale semigroups

Abstract

In this paper we will deal with Lipschitz continuous perturbations of Morse-Smale semigroups with only equilibrium points as critical elements. We study the behavior of the structure of equilibrium points and their connections when subjected to non-differentiable perturbations. To this end we define more general notions of hyperbolicity and transversality, which do not require differentiability.

Continuity and topological structural stability for nonautonomous random attractors

“…This notion of dynamically gradient was studied for random dynamical systems in [23,35]. For topological structural stability of deterministic autonomous or nonautonomous dynamical systems, see [1,10,27]. Now, we present our result on the topological structural stability for random dynamical systems.…”

“…We refer to this property as topological structural stability. Gradient dynamical systems were widely studied in the past years, see [1,10,11,12,30,18] for deterministic dynamical systems, and [23,35] for random dynamical systems. In this work, we obtain a result on the topological structural stability for nonautonomous random differential equations, see Theorem 6.2.…”

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

“…Remark 3.14. In the parabolic case, when −A is sectorial, with A ∈ L(X δ , X), 0 < δ < 1, where X δ is a fractional power of X, we cannot assume that the nonlinearity f 0 : U ⊂ X → X is differentiable, see [10]. We have to assume that the hyperbolic equilibrium y * 0 is in X δ and that U is a open neighborhood of y * 0 in X δ such that f 0 : U ⊂ X δ → X is differentiable with derivative f (y * 0 ) ∈ L(X δ , X).…”

“…Since the perturbation B (t, θ t ω) := [κ t − κ t ]z * (θ t ω) is uniformly bounded on time, we can develop a theory of exponential dichotomies to guarantee existence of hyperbolicity for (1.9). Historically, to study permanence of properties under perturbations it is sensible to assume that the perturbation is uniformly bounded on time, see [2,3,4,6,10,19,23]. For instance, ifẏ = Ay is hyperbolic, and B : R → L(X) is uniformly bounded with respect of t, then the hyperbolicity persists onẏ = Ay + B(t)y, see [23].…”

“…In [19] they study existence and continuity of hyperbolic solutions, invariant manifolds, and provide conditions to prove continuity of pullback attractors for evolution processes. This work was crucial for further development on continuity and stability of attractors in a deterministic scenario [10,11,17]. However, for random dynamical systems the theory is still under development.…”