In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial de...In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings 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 theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.展开更多
基金Project supported by the National Natural Science Foundation of China(No.10171044)the Foundation for University Key Teachers of the Ministry of Education 34C05,34C15,58F14,58F30
文摘In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings 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 theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.
