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

Abstract

In this paper we investigate critical periods for a planar cubicdifferential system with a periodic annulus linking to equilibriaat infinity. The monotonicity of the period function is decided bythe sign of the second order derivative of a Abelian integral. Wederive a Picard-Fuchs equation from a system of Abelian integralsand further give an induced Riccati equation for a ratio ofderivatives of Abelian integrals. The number of critical points ofthe period function for periodic annulus is determined bydiscussing an planar autonomous system, the orbits of whichdescribe solutions of the Riccati equation.