Heteroclinic cycles in dynamical systems with broken spherical symmetry

Abstract

In mathematics and physics the phrase “symmetry breaking” has two distinct meanings. The first refers to the frequently observed phenomenon that a configuration of a physical system satisfying a law (a set of equations) which is invariant under a group of transformations, may itself only be invariant under a subgroup of this group. This is referred to as spontaneous symmetry breaking. The second meaning refers to the process of explicitly adding symmetry breaking terms to the equations which describe the system. This we call induced or forced symmetry breaking. Over the past decade spontaneous symmetry breaking has attracted a considerable amount of attention within the context of bifurcation and dynamical systems theory (see, for example, [13]). Previous work on forced symmetry breaking has mainly concentrated on the persistance of equilibrium solutions under small symmetry breaking perturbations of symmetric systems [3, 4, 2, 7, 8, 12, 22, 231. The main aim of this paper is to show that such perturbations may also give rise to more complex dynamical behavior, in particular to heteroclinic cycles. We first present a simple example of our constructions (for the particular choices of groups G = SO(3), H = SO(2), and K = T, described below) in order to illustrate these general ideas, without too many technicalities; we return to a rigorous discussion of these types of problems in Section 2.