Characterizing  asymptotic stability with Dulac functions

Marc Chamberland; Anna Cima; Armengol Gasull; Francesc Mañosas

doi:10.3934/dcds.2007.17.59

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2007, Volume 17, Issue 1: 59-76. Doi: 10.3934/dcds.2007.17.59

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Characterizing asymptotic stability with Dulac functions

1.
Department of Mathematics and Statistics, Grinnell College, Grinnell, IA, 50112
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra

Received: February 28, 2006

Revised: June 30, 2006

Published: December 2006

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Abstract

This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in $\mathbb{R}^n$. It is well-known that such stability can be characterized in terms of Liapunov functions. The authors prove similar results for the more geometrically motivated Dulac functions. In particular it holds that any analytic autonomous ordinary differential equation having a critical point which is a global attractor admits a Dulac function. These results can be used to give criteria of global attraction in two-dimensional systems.

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Mathematics Subject Classification: Primary: 37C75; Secondary: 34D23, 37C10.

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