Fourth order resonant collisions of multipliers in reversible maps: period-4 orbits and invariant curves

Abstract

Resonant collision of multipliers at ±i of a symmetric fixed point for a 2-parameter family of 4-dimensional reversible maps is considered. Bifurcation of period-4 orbits from the fixed point and their linear stability characteristics are briefly reviewed. In one of the three possible types of bifurcation (see text), a small angle secondary collision of the Floquet multipliers of the bifurcating periodic orbit takes place, leading to the bifurcation of invariant curves from the orbit. The invariant curves are calculated in a perturbation scheme in the leading order of perturbation. The secondary bifurcation is found to be of superthreshold type. An interesting pattern in the vicinity of the resonant collision, involving families of invariant curves and 2-tori, emerges. Results of numerical iterations, corroborating the picture conjectured on the basis of perturbation calculations, are presented. Corresponding results on resonant collisions at −1 are briefly stated.