Approximation Dynamics and the Stability of Invariant Sets

Abstract

We introduce the concept of a weakly, normally hyperbolic set for a system of ordinary differential equations. This concept includes the notion of a hyperbolic flow, as well as that of a normally hyperbolic invariant manifold. Moreover, it has the property that it is closed under finite set products. Consequently, the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly, normally hyperbolic sets. Our main objective is to show that under a small C 1 -perturbation, a weakly, normally hyperbolic set K is preserved by a homeomorphism, where the image K Y is a compact invariant set, with a related hyperbolic structure, for the perturbed equation. In addition, the homeomorphism is close to the identity in C 0, 1 and the perturbed dynamics on K Y are close to the original dynamics on K.1998 Academic Press

Perturbations of Normally Hyperbolic Manifolds with Applications to the Navier–Stokes Equations

“…Consequently, these estimates and (6.41) imply that there are ; 4 , ; 5 , ; 6 # 7 such that ; 5 ; 6 and 58) for 0 t 2T. It follows from inequality (3.9) that, for t 0, one has …”

Perturbations of foliated bundles and evolutionary equations

“…In this paper, we present two important extensions of the basic theory of foliated invariant sets, as developed in [20,21,23,24]. First we extend the perturbation theory to include foliated invariant sets in an infinite-dimensional setting, including the Navier-Stokes equations.…”

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In two earlier papers, we presented a perturbation theory for laminated, or foliated, invariant sets K o for a given finite-dimensional system of ordinary differential equations, see [20,21]. The main objective in that perturbation theory is to show that: if the given vector field has a suitable exponential trichotomy on K o , then any perturbed system that is C 1 -close to the given vector field near K o has an invariant set K n , where K n is homeomorphic to K o and where the perturbed vector field has an exponential trichotomy on K n .

In this paper we present a dual-faceted extension of this perturbation theory to include: (1) a class of infinite-dimensional evolutionary equations that arise in the study of reaction diffusion equations and the Navier-Stokes equations and (2) nonautonomous evolutionary equations in both finite and infinite dimensions.

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“…In two earlier papers, we presented a perturbation theory for laminated, or foliated, invariant sets K o for a given finite-dimensional system of ordinary differential equations, see [20,21]. The main objective in that perturbation theory is to show that: if the given vector field has a suitable exponential trichotomy on K o , then any perturbed system that is C 1 -close to the given vector field near K o has an invariant set K n , where K n is homeomorphic to K o and where the perturbed vector field has an exponential trichotomy on K n .…”

Center manifolds for smooth invariant manifolds

“…Later, it was shown by Mañé ( [34]), and by Bronstein and Kopanskii ( [6]) that, if Y is a C r invariant closed manifold of a C r flow, then r-normal hyperbolicity is equivalent to C r persistence and isolation. Recently, Pliss and Sell ( [40]) introduced the concept of a weakly, normally hyperbolic invariant set and showed persistence results for such a set.…”