Spatio-temporal symmetries inSN- andSN×2-equivariant systems

This paper investigates the spatio-temporal symmetries of periodic trajectories in dynamical systems with S N and S N × 2 symmetry. It turns out that trajectories in S N -equivariant systems cannot exhibit spatio-temporal symmetries beyond the trivial symmetry of all periodic orbits. More complex symmetries in the trajectories require additional constraints on the dynamics. The possibilities offered by S N × 2 symmetric systems are considered and a specific S 3 × 2-equivariant system is investigated numerically.