Fig. 21

A robust heteroclinic cycle attractor for the all-to-all coupled 4-oscillator system (32) with phase interaction function (39) and an open region of parameter space, as in [232]. The figure shows the cycle in the three-dimensional space of phase differences; the vertices R all represent the fully symmetric (inphase) oscillations \((\varphi ,\varphi ,\varphi,\varphi)\), varying by 2π in one of the arguments. The point Q represents solutions \((\varphi,\varphi,\varphi+\pi ,\varphi+\pi )\) with symmetry \((S_{2}\times S_{2})\times_{s} \mathbb {Z}_{2}\). The heteroclinic cycle links two saddle equilibria \(P_{1}=(\varphi,\varphi,\varphi+\alpha ,\varphi +\alpha)\) and \(P_{2}=(\varphi,\varphi,\varphi+2\pi-\alpha,\varphi +2\pi -\alpha)\) with \(S_{2}\times S_{2}\) isotropy. The robust connections \(G_{1}\) and \(G_{2}\) shown in red lie within two-dimensional invariant subspaces with isotropy \(S_{2}\) while the equilibria S have isotropy \(S_{3}\). This structure is an attractor for parameters in the region indicated in Fig. 15