Periodic Solutions of vdP and vdP-like Systems on $3$--Tori

TL;DR

This paper investigates the existence and symmetry properties of periodic solutions in networks of coupled van der Pol and van der Pol-like oscillators arranged on a 3-torus, using equivariant bifurcation and degree theory.

Contribution

It analyzes symmetric periodic solutions in coupled vdP/vdPl systems on a 3-torus, applying equivariant bifurcation theory to both symmetric and non-symmetric coupling scenarios.

Findings

Existence of symmetric periodic solutions via Hopf bifurcation

Global periodic solutions with prescribed period and symmetry

Application of equivariant degree and singularity theory methods

Abstract

Van der Pol equation (in short, vdP) as well as many its non-symmetric generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl depend on symmetries of coupling. In this paper, we consider $N^3$ identical vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the same way as for a $3$-torus. Depending on which nodes impact the dynamics of a given node, we distinguish between $\mathbb D_N \times \mathbb D_N \times \mathbb D_N$-equivariant systems and their $\mathbb Z_N \times \mathbb Z_N \times \mathbb Z_N$-equivariant counterparts. In both settings, the local equivariant Hopf bifurcation together with the global existence of periodic solutions with prescribed period and symmetry, are studied. The methods used in the paper are based on the results rooted in both equivariant degree theory and (equivariant) singularity theory