Stability of heteroclinic cycles

A heteroclinic orbit is a solution of a dynamical system which converges to one stationary solution in the past and to another stationary solution in the future. A heteroclinic chain is a sequence of heteroclinic orbits where the past limit of each orbit is the future limit of the preceding one. If this sequence is periodic we get what is called a heteroclinic cycle. Given such an object it is of interest to ask about its stability. For an initial datum sufficiently close to the cycle, when does the corresponding solution converge to the cycle at late times? In particular, when is the $\omega$-limit set of the solution of interest equal to the entire cycle? To obtain information about this question it is useful to consider the linearization of the system about the vertices of the cycle. For a solution of the kind we are looking for, if it exists, will spend most of its time near the stationary points which are the vertices. If during time periods near a vertex it tends to approach the cycle then this is a good sign that the whole solution will approach the cycle. The behaviour of the solution near the stationary solution is determined by the linearization, at least if the stationary solution is hyperbolic.

In a previous post I described a result of Stefan Liebscher and collaborators which provides detailed information on the nature of the initial singularities of some spatially homogeneous spacetimes which are vacuum or where the matter content is described by a perfect fluid with linear equation of state $p=(\gamma-1)\rho$. In that situation the Einstein equations can be reduced to a system of ordinary differential equations, the Wainwright-Hsu system, which treats all Bianchi class A models in a unified way. In particular it includes the type IX models. There is a heteroclinic cycle consisting of three Bianchi type I vacuum solutions. The main theorem of the paper is that there is a codimension one submanifold of initial data for which the $\alpha$-limit set of the corresponding solution is the heteroclinic cycle just described. The qualitative nature of this result is just as in the general discussion above except that the direction of time has been reversed. The system for vacuum solutions is four-dimensional. The vertices of the cycle are embedded in a one-dimensional manifold of stationary solutions and so the linearization must have at least one zero eigenvalue. As a consequence these vertices are not hyperbolic but the problem can be overcome. Of the remaining eigenvalues one $-\mu$ is negative and the others $\lambda_1,\lambda_2$ are positive. The theorem makes use of the fact that $\lambda_i>\mu$ for $i=1,2$. In the presence of a fluid there is additional positive eigenvalue $\lambda_3$. The same idea of proof applies provided $\lambda_3>\mu$. This inequality is equivalent to an inequality for the parameter $\gamma$ in the equation of state of the fluid.

An analogue of the vacuum solutions of type IX is given by solutions of type ${\rm VI}{}_0$ with a magnetic field. The dynamics of these solutions near the singularity was studied a long time ago by Marsha Weaver. In this situation there is a heteroclinic cycle essentially identical to that in the vacuum case. It is then natural to ask whether an analogue of the known theorem in the vacuum case in the paper by Stefan and collaborators holds. Together with Stefan and Blaise Tchapnda we have now written a paper on this subject. It turns out that there is a closely analogous result but that it is a lot harder to prove. The reason is that the eigenvalues of the linearization are in a less favourable configuration. Fortunately a weaker condition on the eigenvalues suffices. Suppose that $\lambda_1$ denotes the eigenvalue at a vertex corresponding to the outgoing orbit in the cycle at that point. Then it suffices to assume that $\lambda_1>\mu$ without imposing conditions of the other $\lambda_i$, provided that another condition on the existence of invariant manifolds is satisfied. The existence of these manifolds is a consequence of the geometric nature of the problem which gives rise to the dynamical system being considered. In this way we get a result on the stability of the heteroclinic cycle in the model with magnetic field. We are also able to remove the undesirable restriction on $\gamma$ in the case with fluid. This work gives rise to a number of new questions on possible generalizations of this result. For more information on this I refer to the discussion section of our paper.