Resonance bifurcations from robust homoclinic cycles

TL;DR

This paper analyzes resonance bifurcations from robust homoclinic cycles with symmetry, providing optimal stability conditions, confirming supercritical bifurcation nature, and deriving explicit period formulas with strong numerical agreement.

Contribution

It introduces explicit calculations for stability and bifurcation criticality in symmetric homoclinic cycles, resolving a conjecture and deriving parameter-free period formulas.

Findings

Resonance bifurcations are generically supercritical.

Optimal stability conditions are derived using transition matrix techniques.

Explicit asymptotic period formulas match numerical results.

Abstract

We present two calculations for a class of robust homoclinic cycles with symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic stability given by Krupa and Melbourne are not optimal. Firstly, we compute optimal conditions for asymptotic stability using transition matrix techniques which make explicit use of the geometry of the group action. Secondly, through an explicit computation of the global parts of the Poincare map near the cycle we show that, generically, the resonance bifurcations from the cycles are supercritical: a unique branch of asymptotically stable period orbits emerges from the resonance bifurcation and exists for coefficient values where the cycle has lost stability. This calculation is the first to explicitly compute the criticality of a resonance bifurcation, and answers a conjecture of Field and Swift in a particular limiting case. Moreover, we are able to obtain an asymptotically-correct analytic expression for the period of the bifurcating orbit, with no adjustable parameters, which has not proved possible previously. We show that the asymptotic analysis compares very favourably with numerical results.