Codimension 2 Symmetric Homoclinic Bifurcations and Application to 1:2 Resonance

Abstract

In this paper we study a codimension 3 form of the 1:2 resonance. It was first noted by Arnold [3] that the study of bifurcations of symmetric vector fields under a rotation of order q yields information about Hopf bifurcation for a fixed point of a planar diffeomorphism F with eigenvalues . The map Fq can be identified to arbitrarily high order with the flow map of a symmetric vector field having a double-zero eigenvalue ([3], [4], [10], [23]). The resonance of order 2 (also called 1:2 resonance) considered here is the case of a pair of eigenvalues —1 with a Jordan block of order 2. The diffeomorphism then has normal form around the origin given by [4]: