An example of a non-asymptotically stable attractor

Nonlinearity 4 (1991) 835-844

Ian Melbourne


We give an example of an invariant set that is not asymptotically stable but which has the following strong attracting properties. `Almost all' trajectories that start close to the invariant set behave as if the set were asymptotically stable, that is, these trajectories remain close and converge to the invariant set. The term `almost all' means here that the only trajectories that escape lie in a cuspoidal region abutting the invariant set.

Our example is a heteroclinic cycle forced by symmetry. The surprising feature is that nodes on the cycle may have unstable eigenvalues in directions `normal' to the cycle, and yet the cycle is stable in the above sense. This type of stability appears to explain some numerical experiments.