Bifurcations of coupled nonlinear oscillators with similar kinematics

Abstract

The application of the principle of skew symmetry for nonlinear systems that represent a bunch of nonlinear Van der Pol oscillators is analyzed. A bunch of oscillators can (depending on the parameters) form systems of coupled regular limiting cycles and coupled attractors with chaotic or conditionally periodic winding of the trajectory. At a slight change in the parameters of oscillators, the scale of two limiting cycles changes. A strong change in the parameters and the coupling coefficient leads to the appearance of limiting cycles with chaotic winding of the trajectory. When considering three connected limiting cycles, one can reduce them to two ones with a periodic winding and one limiting cycle with a conditionally periodic winding. To clarify the nature of the winding of the trajectories, a topological analysis of the trajectory should be done. In this case, the equations in variations are constructed, and the characteristic indicators of solutions are found.