Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry

Proc. Roy. Soc. Edinburgh 113A (1989) 315-345
Also in New Directions in Differential Equations and Dynamical Systems J.M. Ball ed. Roy. Soc. Edinburgh (1991)

Ian Melbourne, Pascal Chossat and Marty Golubitsky


In this paper we show that in O(2) symmetric systems, structurally stable, asymptotically stable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions.

We find these cycles using group-theoretic techniques; in particular, we look for certain patterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.