Dynamical analysis of a cubic Liénard system with global parameters3Supported by NSFC 11471228, 11172246 and the Fundamental Research Funds for the Central Universities. (4th September 2015) - Chen, Hebai; Chen, Xingwu
Abstract: In this paper we investigate the dynamical behaviour of a cubic Liénard system with global parameters. After analysing the qualitative properties of all the equilibria and judging the existences of limit cycles and homoclinic loops for the whole parameter plane, we give the bifurcation diagram and phase portraits. Phase portraits are global if there exist limit cycles and local otherwise. We prove that parameters lie in a connected region, not just on a curve, usually in the parameter plane when the system has one homoclinic loop. Moreover, for global parameters we give a positive answer to conjecture 3.2 of (1998 Nonlinearity 11 1505–19) in the case of exactly two equilibria about the existence of some function whose graph is exactly the surface of double limit cycles.