On finite-time hyperbolicity

Arno Berger

doi:10.3934/cpaa.2011.10.963

Article Contents

2011, Volume 10, Issue 3: 963-981. Doi: 10.3934/cpaa.2011.10.963

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On finite-time hyperbolicity

Arno Berger^1,

1.
Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1

Received: March 2009

Revised: September 2009

Published: May 2011

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Abstract

A solution of a nonautonomous ordinary differential equation is finite-time hyperbolic, i.e. hyperbolic on a compact interval of time, if the linearisation along that solution exhibits a strong exponential dichotomy. As a finite-time variant and strengthening of classical asymptotic facts, it is shown that finite-time hyperbolicity guarantees the existence of stable and unstable manifolds of the appropriate dimensions. Eigenvalues and -vectors are often unsuitable for detecting hyperbolicity. A (dynamic) partition of the extended phase space is used to circumvent this difficulty. It is proved that any solution staying clear of the elliptic and degenerate parts of the partition is finite-time hyperbolic. This extends and unifies earlier partial results.

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Mathematics Subject Classification: Primary: 34A30, 37B55, 37D05; Secondary: 34D09, 37D10, 37N10.

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