Bolotov D. I., Bolotov M. I., Smirnov L. A., Osipov G. V., Pikovsky A.
Twisted States in a System of Nonlinearly Coupled Phase Oscillators
2019, vol. 24, no. 6, pp. 717-724
We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott – Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.
Grines E. A., Osipov G. V.
Heteroclinic and Homoclinic Structures in the System of Four Identical Globally Coupled Phase Oscillators with Nonpairwise Interactions
2018, vol. 23, no. 7-8, pp. 974-982
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviors. Such systems can have chaotic dynamics for $N > 4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies have shown that some of chaotic attractors in this system are organized by heteroclinic networks. In the present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddle-foci equilibria.