Dynamics of two van der Pol oscillators coupled via a bath

In this work we study a system of two van der Pol oscillators, x and y, coupled via a "bath" z: ẍ-ε(1-x²)ẋ+x=k(z-x) ÿ-ε(1-y²)ẏ+y=k(z-y) ż=k(x-z)+k(y-z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε>0 and k>0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. In our model although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath.