Synchronization of Van der Pol oscillators in a thermal bath

The phenomenon of synchronization in self-sustained systems has been successfully illuminated in many fields, ranging from biology to electrical engineering. To date, the majority of theoretical studies on synchronization focus on isolated self-sustained systems, leaving the effects of surrounding environments less touched due to the lack of appropriate descriptions. Here we derive a generalized Langevin equation that governs the dynamics of open classical Van der Pol (VdP) oscillators immersed in a common thermal bath with arbitrary memory time and subsumes an existing equation for memoryless bath as a special limit. The so-obtained Langevin equation reveals that the bath can induce a dissipative coupling between VdP oscillators, besides the usual damping and thermal noise terms connected by the fluctuation-dissipation theorem. To demonstrate the utility of the approach, we investigate a model system consisting of two open VdP oscillators coupled to a thermal bath with an Ohmic or a Lorentzian-shape spectrum. Unlike the isolated setup where the stable synchronization can be either in-phase or antiphase when varying initial conditions, we find that the bath always favors a single type of synchronization in the long-time limit regardless of initial conditions and the synchronization type can be switched by tuning the temperature. Moreover, we show that the bath-induced dissipative coupling can trigger a synchronization of open VdP oscillators that is otherwise absent between isolated counterparts. Our results complement and extend previous findings for open VdP oscillators.