Resonant Collisions in Four Dimensional Reversible Maps: A Description of Scenarios

Abstract

We define a resonant collision of order k(geq 1) in a family of four-dimensional reversible maps. For any specified k, the bifurcation secnario is the collection of the different possible types of bifurcation of a symmetric fixed point that may be encountered through various choices of parameters describing the family of maps under consideration. We adopt a perturbative approach, coupled with numerical iterations around orbits obtained perturbatively, to explore phase space structures in the immediate vicinity of a resonant collision and thereby to obtain a description of the possible scenarios for different values of k. The phase space structures typically involve bifurcating periodic orbits, families of invariant curves, and tori, and present interesting possibiliries, especially around the 'secondary bifurcations' of the periodic orbits (see below). Based on the results of the perturbative and numerical approach we conjecture that three distinct scenarios are involved for the cases k = 2, 3, 4 respectively, while there exists a fourth distinctive scenario common to all k > 4, and we present what we believe to be a reasonably exhaustive description of these scenarios. The case k = 1 involves bifurcations of fixed points rather than of periodic orbits, and has been investigated numerically in a previous paper.