On asymptotic phase of dynamical system hyperbolic along attracting invariant manifold

TL;DR

This paper investigates the asymptotic phase property of dynamical systems with hyperbolic structure near an attracting invariant manifold, demonstrating that previous decay rate conditions are unnecessary for the existence of asymptotic phases.

Contribution

It proves that the decay rate condition is not needed for asymptotic phase existence, using Brouwer's fixed point theorem and analyzing invariant foliation structures.

Findings

Neighborhoods are filled with motions having asymptotic phases

Decay rate condition is not necessary for asymptotic phase existence

Invariant foliation structure is characterized near the manifold

Abstract

We consider a dynamical system which has the hyperbolic structure along an attracting invariant manifold $M$. The problem is whether every motion starting in a neighborhood of $M$ possesses an asymptotic phase, i.e. eventually approaches a particular motion on $M$. Earlier, positive solutions to the problem were obtained under the condition that the decay rate of solutions toward the manifold exceeds the decay rate of the solutions within the manifold. We show that in our case the above condition is not necessary. To prove that a neighborhood of $M$ is filled with motions for each of which there exists an asymptotic phase we apply the Brouwer fixed point theorem. An invariant foliation structure which appears in the neighborhood of $M$ is discussed.