On local attraction properties and a stability index for heteroclinic connections

Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set $X\subset\R^n$ with a basin of attraction $N$, we define a stability index $\sigma(x)$ of a point $x\in X$ that characterizes the local extent of the basin. Let $B_{\epsilon}$ denote a ball of radius $\epsilon$ about $x$. If $\sigma(x)>0$, then the measure of $B_{\epsilon}\setminus N$ relative the measure of the ball is $O(\epsilon^{|\sigma(x)|})$, while if $\sigma(x)<0$, then the measure of $B_{\epsilon}\cap N$ relative the measure of the ball is of the same order. We show that this index is constant along trajectories, and relate this orbit invariant to other notions of stability such as Milnor attraction, essential asymptotic stability and asymptotic stability relative to a positive measure set. We adapt the definition to local basins of attraction (i.e. where $N$ is defined as the set of initial conditions that are in the basin and whose trajectories remain local to $X$). This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in $\R^4$ and show that the local stability indices (and hence local stability properties) can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle. In doing this, we extend previous results of Krupa and Melbourne (1995,2004) and give criteria for simple heteroclinic cycles in $\R^4$ to be Milnor attractors.