Symmetric solutions to a system of mutually delay-coupled oscillators with conjugate coupling

Abstract

Coupled oscillators are the subject of different studies because they display interesting behavior, such as synchrony. In this paper, we investigate the dynamics of a system of two delay-coupled oscillators and show that they display some kind of synchrony. We use the Stuart-Landau equations to represent this system, which is linked via conjugate coupling. These equations form a system of delay differential equations, which we have found to have symmetry isomorphic to Z2 × Z2. We looked into the form of steady-states that inherit part of the symmetry of this system and the birth of periodic solutions with symmetry. In particular, we found bifurcating steady-state and periodic solutions with symmetry isomorphic to Z2 emanating from the branch of trivial solutions. As an illustration, we performed numerical continuation to generate a bifurcating branch of symmetric solutions from the branch of trivial solutions. These symmetric solutions provide us with the type of synchrony that can be displayed by the oscillators. Finally, we determined the stability of the different branches of symmetric solutions numerically.