Homoclinic explosion in the vicinity of bifurcation at the triple-zero eigenvalue

Abstract

Equations describing small-amplitude motion in the proximity of bifurcation at the triple-zero eigenvalue, after being reduced to a normal form by nonlinear transformations, display a double-scale structure that combines relatively fast conservative orbital motion with slow dissipative evolution of two integrals of motion. The homoclinic explosion occurs when the attractor of the averaged dissipative equations comes close to a homoclinic trajectory of the fast conservative subsystem. Applying the method of matched asymptotic expansions allows one to reduce the dynamical system near this point to a simple next return map possessing an infinite family of invariant manifolds that can turn into attractors of different character under small parametric perturbation. After the attractor of the averaged system has been expelled from the region of closed orbits, one observes large-amplitude dynamics with rare events of anomalous intermittency superimposed upon a quasiperiodic or chaotic attractor of nonuniversal character.