Versal unfolding of a nilpotent Liénard equilibrium within the odd Liénard family

Abstract

Recently attentions were paid to versal unfolding of degenerate equilibria within a restricted family of vector fields for a practical sense. The family of Liénard systems is such a family of significant physical sense but only within the family of even Liénard systems there were found results for a nilpotent degenerate equilibrium. In this paper we discuss versal unfolding of a nilpotent Liénard equilibrium within the family of odd Liénard systems. As the well-known Bogdanov-Takens normal form is not available for the odevity, we prove that the degeneracy within the family is of codimension 2 and find its versal unfolding. We further discuss the unfolding system, displaying all its bifurcations such as pitchfork bifurcation, center-saddle bifurcation, center bifurcation and homoclinic (heteroclinic) loop bifurcation, which show how a single homoclinic loop, a twin homoclinic loop and a heteroclinic loop arise from a center or a (degenerate) saddle.