Singular perturbations of finite dimensional gradient flows

Chiara Zanini

doi:10.3934/dcds.2007.18.657

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2007, Volume 18, Issue 4: 657-675. Doi: 10.3934/dcds.2007.18.657

This issue Previous Article Weak geodesic flow and global solutions of the Hunter-Saxton equation Next Article Action functionals that attain regular minima in presence of energy gaps

Singular perturbations of finite dimensional gradient flows

Chiara Zanini^1,

1.
SISSA, via Beirut 4, 34014 Trieste

Received: August 31, 2006

Revised: November 30, 2006

Published: September 2007

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Abstract

In this paper we give a description of the asymptotic behavior, as $\varepsilon\to 0$, of the $\varepsilon$-gradient flow in the finite dimensional case. Under very general assumptions, we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).

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Mathematics Subject Classification: 34K26, 34K18, 34C37.

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