Synchronization dynamics and numerical simulation of three coupled oscillators through extended Poincaré-Cartan invariants

This article presents the numerical simulation of three linearly coupled oscillators, which are connected to form the circadian rhythms for analysing the motion of eyes. In this system of circadian rhythms, each eye is modeled as van der Pol oscillators and both such oscillators are not directly interconnected but connected through another similar oscillator. This oscillator models the pineal gland, which is responsible for linkage with brain muscles. The van der Pol oscillators are coupled with such couplings, where non-linear damping parameters are also included in this system. The dynamics is determined analytically through a novel approach of extended Poincaré-Cartan Invariants of Lagrangian theory, which demonstrates the invariance of Poincare integral in three-dimensional phase space. The synchronization dynamics is achieved with in-phase and out of phase motions of oscillators with varying parameters. In this analysis, the effect of injected signal is also analyzed by adding some detuning parameter considering the associated boundaries of the stable synchronized states. This analysis is corroborated with simulation results to visualize the dynamical behaviour.