Abstract
Continuing Chen and Chen (2015 Nonlinearity 28 3535) and (2016 Nonlinearity 29 1798) which deal with the cases of two equilibria and three equilibria respectively, in this paper we investigate the global dynamics of a cubic Liénard system with global parameters in the case of exact one equilibrium. After analyzing qualitative properties of all equilibria and judging the number of limit cycles, we give the bifurcation diagram and all global phase portraits. Our method in judging the number of limit cycles is to construct a parameter transformation such that in new parameter space the vector field is rotated about multiple parameters and, hence, is essential different from the methods used in previous publications. Associated with the results of last two publications, we get a positive answer to conjecture 3.2 of Khibnik et al (1998 Nonlinearity 11 1505) for general parameters about the existence of some function whose graph is exactly the surface of the double limit cycle bifurcation and therefore solve this conjecture completely.
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