One-way coupled Van der Pol system

TL;DR

This paper introduces a one-way coupled Van der Pol system that is reformulated within an action principle framework, revealing new bifurcation and synchronization behaviors in chaotic regimes.

Contribution

It presents a novel formulation of the Van der Pol oscillator as a one-way coupled system within an action principle, analyzing its bifurcations and synchronization in chaos.

Findings

System exhibits bifurcations different from uncoupled Van der Pol oscillator.

System is chaotic with phase synchronization observed at small μ.

Critical points depend on the parameter μ, affecting stability.

Abstract

The equation of the Van der Pol oscillator, being characterized by a dissipative term, is non-Lagrangian. Appending an additional degree of freedom we bring the equation in the frame of action principle and thus introduce a one-way coupled system. As with the Van der Pol oscillator, the coupled system also involves only one parameter that controls the dynamics. The response system is described by a linear differential equation coupled nonlinearly to the drive system. In the linear approximation the equations of our coupled system coincide with those of the Bateman dual system (a pair of damped and anti-damped harmonic oscillators). The critical point of damped and anti-damped oscillators are stable and unstable for all physical values of the frictional coefficient $\mu$. Contrarily, the critical points of the drive- (Van der Pol) and response systems depend crucially on the values of $\mu$. These points are unstable for $\mu > 0$ while the critical point of the drive system is stable and that of the response system is unstable for $\mu < 0$. The one-way coupled system exhibits bifurcations which are different from those of the uncoupled Van der Pol oscillator. Our system is chaotic and we observe phase synchronization in the regime of dynamic chaos only for small values of $\mu$.