Evgeny Grines
ul. Gagarina 23, Nizhni Novgorod, 606950 Russia
Nizhni Novgorod State University
Publications:
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, nos. 78, pp. 974982
Abstract
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 saddlefoci equilibria.

Grines E. A., Osipov G. V.
On Constructing Simple Examples of Threedimensional Flows with Multiple Heteroclinic Cycles
2015, vol. 20, no. 6, pp. 679690
Abstract
In this work we suggest a simple method for constructing $G$equivariant systems of ODEs in $\mathbb{R}^3$ (i.e., systems whose trajectories are invariant under the action of this group on $\mathbb{R}^3$) that possess multiple disjoint heteroclinic networks. Heteroclinic networks under consideration consist of saddle equilibria that belong to coordinate axes and onedimensional separatrices connecting them. We require these separatrices to lie on coordinate planes. We also assume the action of $G$ on $\mathbb{R}^3$ to be generated by cyclic permutation of coordinate variables and reflection with respect to one of the coordinate planes. As an example, we provide a stepbystep construction of threedimensional flow with two disjoint heteroclinic networks. Also, we present a sketch of global dynamics analysis for the minimal example.
