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July  2003, 9(4): 1073-1077. doi: 10.3934/dcds.2003.9.1073

Stability of parameterized Morse-Smale gradient-like flows

Ming-Chia Li 1,

1. 

Department of Mathematics, National Changhua University of Education, Changhua 500

Published  April 2003

In this paper, we are concerned with the structural stability of a Morse-Smale gradient-like flow $\varphi^t$ and show that if $\{\varphi_\epsilon^t\}$ is a smooth one-parameter family of $C^3$ flows with $\varphi_0^t=\varphi^t$, and {$\psi_\epsilon^t$} is another one-parameter family of $C^3$ flows such that $\psi_\epsilon^t$ is $C^0$ $O(\epsilon^3)$-close and $C^1$ $O(\epsilon^2)$-close to $\varphi_\epsilon^t$, then for all small $|\epsilon|$, there is a homeomorphism $h_\epsilon$, which is $C^0$ $O(\epsilon^2)$-near the identity map, such that $h_\epsilon$ takes the trajectories of $\varphi^t_\epsilon$ to the ones of $\psi^t_\epsilon$.
Keywords: Morse-Smale gradient flow, averaging method., structural stability, topological conjugacy.
Mathematics Subject Classification: 37D15, 34D30, 37C20, 34C2.
Citation: Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073
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