Critical periods of a periodic annuluslinking to equilibria at infinity in a cubic system

Zhirong He; Weinian Zhang

doi:10.3934/dcds.2009.24.841

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2009, Volume 24, Issue 3: 841-854. Doi: 10.3934/dcds.2009.24.841

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Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system

Zhirong He^1, and
Weinian Zhang^2,

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064
2.
Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received: August 31, 2007

Revised: October 31, 2007

Published: August 2009

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Abstract

In this paper we investigate critical periods for a planar cubic differential system with a periodic annulus linking to equilibria at infinity. The monotonicity of the period function is decided by the sign of the second order derivative of a Abelian integral. We derive a Picard-Fuchs equation from a system of Abelian integrals and further give an induced Riccati equation for a ratio of derivatives of Abelian integrals. The number of critical points of the period function for periodic annulus is determined by discussing an planar autonomous system, the orbits of which describe solutions of the Riccati equation.

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Mathematics Subject Classification: 34C08, 37J45.

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