Explicit and non explicit limit cycles for Liénard equations [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Armengol Gasull
Universitat Autònoma de Barcelona


We consider a family of planar vector fields that writes as a Liénard system in   suitable coordinates. It has an explicit solution that often contains  periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits and we also study the coexistence of them with other non explicit periodic orbits. Our family contains the celebrated Wilson  polynomial Liénard equation, as well as all polynomial Liénard systems having hyperelliptic limit cycles. As an illustrative example we study in more detail a natural 1-parametric extension of Wilson example that has at least two limit cycles, one of them algebraic,  presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. To prove that for some values of the parameter the system has exactly two hyperbolic limit cycles we use several suitable Dulac functions.

This talk is based on a work in progress with M. Sabatini.