DETECTING THE SYMMETRY OF ATTRACTORS FOR SIX OSCILLATORS COUPLED IN A RING

Abstract

We study a dynamical system modelling six coupled identical oscillators, introduced by Collins and Stewart in connection with hexapod gaits. The system has dihedral group symmetry D6, and they suggest that symmetric chaos may be present at some parameter values. We confirm this by employing the method of “detectives” introduced by Barany, Dellnitz, and Golubitsky. The paper is mainly intended as a case study in the use of detectives to analyse the symmetries of attractors, employing a system with a rich and varied range of bifurcations—both changing the symmetry and changing the dynamics—with a reasonably large dimensional phase space (here 12 dimensions). We confirm that detectives provide a quick and effective method for finding parameter values at which symmetry changes or other bifurcations occur. However, because of ambiguities in interpretation in practical cases, they should be supplemented, when necessary, by other diagnostic techniques. Convergence plots for ergodic sums appear to be especially useful, in addition to standard techniques such as phase portraits and Poincaré sections.