Characterizing asymptotic stability with dulac functions

Marc Chamberland; Anna Cima; Armengol Gasull; Francesc Mañosas

Characterizing asymptotic stability with dulac functions

Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

3 Citations (Scopus)

Abstract

This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in ℝn. 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.

Original language	English
Pages (from-to)	59-76
Journal	Discrete and Continuous Dynamical Systems
Volume	17
Issue number	1
Publication status	Published - 1 Jan 2007

Keywords

Curvature of orbits
Dulac function
Global and local asymptotic stability
Jacobian conjecture
Liapunov function
Markus-Yamabe

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title = "Characterizing asymptotic stability with dulac functions",

abstract = "This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in ℝn. 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 = "Curvature of orbits, Dulac function, Global and local asymptotic stability, Jacobian conjecture, Liapunov function, Markus-Yamabe",

author = "Marc Chamberland and Anna Cima and Armengol Gasull and Francesc Ma{\~n}osas",

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AU - Cima, Anna

AU - Gasull, Armengol

AU - Mañosas, Francesc

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N2 - This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in ℝn. 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.

AB - This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in ℝn. 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.

KW - Curvature of orbits

KW - Dulac function

KW - Global and local asymptotic stability

KW - Jacobian conjecture

KW - Liapunov function

KW - Markus-Yamabe

M3 - Article

VL - 17

SP - 59

EP - 76

IS - 1

ER -