C. Hou and M. Golubitsky
An example of symmetry breaking to heteroclinic cycles
J. Diff. Eqn. 133 (1) (1997) 30-48.
Lauterbach and Roberts showed that when symmetry breaking terms are added to an equivariant differential equation with a group orbit of equilibria, heteroclinic cycles connecting equilibria on the perturbed group orbit may result. More precisely, let Gamma contained in O(n) be a Lie group acting on R^n and let dot{z} = f(z) Gamma-equivariant system of differential equations; that is, f(gamma z) = gamma f(z) for all gamma in Gamma. Suppose that this differential equation has an equilibrium at z_0. Then equivariance implies that the manifold X_0 = Gamma z_0 is a group orbit of equilibria. We assume that this group orbit is normally hyperbolic; indeed, we assume that X_0 is orbitally asymptotically stable.
Suppose now that we consider a small system symmetry breaking perturbation dot{z} = f(z) + epsilon g(z) where epsilon is small and g is only Delta-equivariant where Delta contained in Gamma is a Lie subgroup. When epsilon is sufficiently small, normal hyperbolicity guarantees that there is a (perturbed) flow invariant manifold X for the perturbed system which is diffeomorphic to X_0. However, the dynamics of the perturbed flow on X need not consist only of equilibria. Indeed, when dim Delta < dim Gamma the dynamics on the perturbed orbit X will generally be more complicated than just consisting of equilibria. Lauterbach and Roberts show that, depending on the pair Gamma and Delta, certain equilibria may be forced to occur on X. In addition, one-dimensional flow invariant sets connecting these equilibria may be forced by the residual symmetry Delta, so that generically heteroclinic cycles connecting the equilibria on X can be forced. Lauterbach and Roberts give an example where heteroclinic cycles are forced by system symmetry breaking. In this example Gamma=O(3) and Delta=T (the group of symmetries of the tetrahedron). More recently, Lauterbach et al. have classified all pairs Gamma and Delta which may result in heteroclinic cycles when Gamma is either O(3) or SO(3) and Delta is any proper Lie subgroup. This investigation is completed when R^n is any of the natural irreducible representations of O(3).
We follow the work by Hou in his thesis and continue this line of investigation by constructing an example where heteroclinic cycles are forced by system symmetry breaking which is simpler than those of Lauterbach and co-workers. In our example Gamma=D_4 dot{+}R^2, Delta=D_2, and n=4. These groups occur when studying bifurcations of spatially periodic solutions to planar Euclidean equivariant systems of PDEs on a square lattice (see Dionne and Golubitsky).
We use equivariant bifurcation theory in the presence of D_4 dot{+}T^2 symmetry to establish the existence of orbitally stable mixed mode equilibria. That is, we assume that the vector field f depends on a bifurcation parameter lambda dot{z} = f(z,lambda). We show that under certain easily verifiable conditions on the lower order terms of f, there exists a branch of orbitally asymptotically stable mixed mode equilibria. We then show that if the first derivatives of g at the origin satisfies certain inequalities, then for all fixed lambda sufficiently near zero and all sufficiently small epsilon (depending on lambda) the perturbed system has a structurally stable, asymptotically stable, heteroclinic cycle.
Once we have established these conditions, we also show that the existence of this cycle depends only on the existence of the D_4 dot{+}T^2-equivariant bifurcation to mixed mode solutions. That is, there exist an open set of perturbation terms g that force the existence of the desired cycle. We can then use these results to prove the existence of heteroclinic cycles in the dynamics of a reaction-diffusion system.