A Unified Theory for Inertial Manifolds, Saddle Point Property and Exponential Dichotomy

TL;DR

This paper unifies the theories of inertial manifolds, saddle point property, and exponential dichotomy, providing a common framework to analyze the splitting of spaces in dynamical systems, with applications to non-autonomous PDEs.

Contribution

It offers a unified proof connecting inertial manifold theory, saddle point property, and exponential dichotomy, enhancing understanding of space splitting in dynamical systems.

Findings

Unified proof for inertial manifold theorem

Establishment of saddle-point property with invariant manifolds

Demonstration of hyperbolicity of global solutions in non-autonomous PDEs

Abstract

Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of `splitting' the space to understand the dynamics. We present a unified proof for the inertial manifold theorem, which as a local consequence yields the saddle-point property with a fine structure of invariant manifolds and the roughness of exponential dichotomy. In particular, we use these tools in order to establish the hyperbolicity of certain global solutions for non-autonomous parabolic partial differential equations.