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January  2007, 17(1): 59-76. doi: 10.3934/dcds.2007.17.59

Characterizing asymptotic stability with Dulac functions

Marc Chamberland 1, , Anna Cima 2, , Armengol Gasull 2, and  Francesc Mañosas 2,

1. 

Department of Mathematics and Statistics, Grinnell College, Grinnell, IA, 50112, United States
2. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain, Spain, Spain

Received  March 2006 Revised  July 2006 Published  October 2006

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.
Keywords: Global and local asymptotic stability, curvature of orbits., Liapunov function, Markus-Yamabe, Jacobian conjecture, Dulac function.
Mathematics Subject Classification: Primary: 37C75; Secondary: 34D23, 37C1.
Citation: Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas. Characterizing asymptotic stability with Dulac functions. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 59-76. doi: 10.3934/dcds.2007.17.59
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