Dynamics and geometry in forced symmetry breaking: a tetrahedral example

Abstract

We are interested in the dynamics that can arise close to rotational group orbits after forced symmetry-breaking to discrete symmetries. In particular we ask how simple or complicated the dynamics induced by symmetric linear vector fields can be. We look in detail at related tetrahedral and dihedral examples, and there we find precise conditions for a linear field to exhibit homoclinic orbits.

Singularity theory and forced symmetry breaking in equations

“…For instance, a simple example in tune with previous example is the equationü + q(u, λ) + µh(t, u,u) = 0 when h is 2π-periodic in t andü + q(u) = 0 has nontrivial periodic orbits (see [36] and [4]). More generally, there has been more recently interest about the effect of forced symmetry breaking on the dynamics around some special solutions, invariant manifolds (Galante and Rodrigues [17], Chillingworth and Lauterbach [5], Comanici [7], Parker et al [30] to mention but a few). In these papers, somewhat different techniques are used, linked with equivariant differential geometry.…”

Persistence of stationary motion under explicit symmetry breaking perturbation

“…Explicit symmetry breaking is defined as a process of perturbing symmetric dynamical equations so that the resulting equations have a lower symmetry group. When the system is not hamiltonian interesting results are obtained showing for example that periodic solutions of an unperturbed dynamical system can become heteroclinic cycles under a perturbation that breaks the symmetry [CL04,GL01,LR92].…”

Symplectic slice for subgroup actions

“…In the study of explicit symmetry breaking phenomenons, Hamiltonian equations are perturbed in a way that the symmetry group G breaks into one of its subgroup H. This phenomenon is studied by many authors. References for the non-Hamiltonian case are for instance [12] or [3]. Some aspects of the Hamiltonian case are studied in [1], [8], [7] or [32].…”