A three-dimensional singularly perturbed conservative system with symmetry breaking

Abstract

In this paper we present an analysis of a three-dimensional singularly perturbed conservative system. We add a constant vector in the vector field to remove one of the symmetries in the system. Using the geometric argument, and a theorem which is derived from the implicit function theorem, we prove the existence of equilibria in the system and also derive some local bifurcations of these equilibria, i.e. saddle-node bifurcations. We also show that although we have two saddle-nodes in the system, the codimension-2 bifurcation called the cusp bifurcation is not possible. More sophisticated bifurcations, such as the Hopf bifurcation and the bifurcation of the created periodic solution are derived by using the numerical continuation software MATCONT. Following the periodic solution, we found a sequence of period-doubling bifurcations which leads to the existence of infinitely many periodic solutions. The coexistence of two attractors is an interesting phenomenon which is observed in this paper. Using asymptotic analysis, we discuss the dynamics in the neighborhood of a particular line in phase space at which the competition between these attractors takes place.

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