Stability of heteroclinic cycles in transverse bifurcations

Abstract

Heteroclinic cycles and networks exist robustly in dynamical systems with symmetry. They can be asymptotically stable, and gradually lose this stability through a variety of bifurcations, displaying different forms of non-asymptotic stability along the way. We analyse the stability change in a transverse bifurcation for different types of simple cycles in R4. This is done by first showing how stability of the cycle or network as a whole is related to stability indices along its connections — in particular, essential asymptotic stability is equivalent to all local stability indices being positive. Then we study the change of the stability indices. We find that all cycles of types B and C are generically essentially asymptotically stable after a transverse bifurcation, and that no type B cycle can be almost completely unstable (unlike type C cycles).