Fig. 24

a Graph of heteroclinic connections between three cluster states for a robust heteroclinic attractor in a system of \(N=5\) globally coupled phase oscillators. Phase interaction function and parameters as in Fig. 23; see [252] for more details of the dynamics. Each vertex represents a particular saddle periodic cluster state that is a permutation of the states shown in b–d. Note that there are precisely two incoming and two outgoing connections at each vertex. b–d show the relative phases of the five oscillators, indicated by the angles of the “pendula” at the vertices of a regular pentagon, for a sequence of three consecutive three saddle cluster states visited on a longer trajectory that randomly explores graph a in the presence of noise. Inequivalent clusters are characterised by different coloured “pendula” and numbers where yellow corresponds to 1, green to 2 and blue to 3. The yellow cluster is stable to cluster-splitting perturbations while the blue cluster is unstable to such perturbations—observe that after the connection the yellow cluster becomes the blue cluster while the blue cluster splits up in one of two possible ways