Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three–fold Zero Eigenvalue

摘要

In this paper we study the singularity at the origin with three–fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent to y(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax~2y (partial deriv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity by using the normal form and blow–up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Silnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic or non–hyperbolic equilibrium in a highdimensional space.