I. Melbourne, P. Chossat and M. Golubitsky
Heteroclinic cycles involving periodic solutions in mode
interactions with 0(2) symmetry
Proc. Roy. Soc. Edinburgh 113A (1989) 315-345.
In this paper we show that in 0(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.