Algebraic approximations to bifurcation curves of limit cycles for the Liénard equation

In this paper, we study the bifurcation of limit cycles in Liénard systems of the form dx dt = y − F (x) , dy dt = −x , where F (x) is an odd polynomial that contains, in general, several free parameters. By using a method introduced in a previous paper, we obtain a sequence of algebraic approximations to the bifurcation sets, in the parameter space. Each algebraic approximation represents an exact lower bound to the bifurcation set. This sequence seems to converge to the exact bifurcation set of the system. The method is non perturbative. It is not necessary to have a small or a large parameter in order to obtain these results. PACS numbers : 05.45.+b , 02.30.Hq , 02.60.Lj , 03.20.+i