Mathematical Proceedings of the Cambridge Philosophical Society

Group theoretic conditions for existence of robust relative homoclinic trajectories

a1 School of Mathematical Sciences, Laver Building, University of Exeter, Exeter EX4 4QE.
a2 CNRS-INLN, 1361 route des Lucioles, 06560 Valbonne, France.


We consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these results we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.

(Received November 14 2000)
(Revised June 6 2001)