# American Institute of Mathematical Sciences

November  2007, 18(4): 657-675. doi: 10.3934/dcds.2007.18.657

## Singular perturbations of finite dimensional gradient flows

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Received  September 2006 Revised  December 2006 Published  May 2007

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).
Citation: Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657
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