The dynamics of a kind of Liénard system with sixth degree and its limit cycle bifurcations under perturbations (Q2301834)

The different topological types of phase portraits of a class of unperturbed Liénard systems are given. The authors note that the expansion of the Melnikov function near any closed orbit appearing in the above phase portraits, except a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle of order one, has been studied. In this paper, the authors give the expansion of the Melnikov function near this kind of heteroclinic loop. Further, they present the conditions to obtain limit cycles bifurcated from a compound loop with a hyperbolic saddle and a nilpotent saddle of order one, and apply it to study the number of limit cycles for a class of Liénard systems under perturbations. The obtained results enrich the bifurcation theory of differential systems to some degree.