We present a perturbation solution for a linear oscillator with a variable damping coefficient involving the limit cycle of the van der Pol equation (van der Pol 1926). This equation arises as the variational equation governing the stability of in-phase vibration in a pair of identical van der Pol oscillators with linear coupling. The van der Pol oscillator has served as the classic example of a limit cycle oscillator, and coupled limit cycle oscillators appear in mathematical models of self-excited systems ranging from tube rows in cross flow heat exchangers to arrays of stomates in plant leaves.
As in many systems modeled by coupled oscillators, criteria for phase-locking or synchronization are of fundamental importance in understanding the dynamics. In this paper we study a simple but interesting problem consisting of a pair of identical van der Pol oscillators with linear diffusive coupling which corresponds, in the mechanical analogy, to a spring connecting the masses of the two oscillators. Intuition and earlier first-order analyses suggest that the spring will pull the two masses together causing stable in-phase locking. However, previous results of a relaxation limit study (Storti and Rand 1986) indicate that the in-phase mode is not always stable and suggest the existence of an additional stability boundary.
To resolve the apparent discrepancy, we obtain a new periodic solution of the variational equation as a power series in ε, the small parameter in the sinusoidal van de Pol oscillator. This approach follows Andersen and Geer’s (1982) solution for the limit cycle of an isolated van der Pol oscillator. The coupling strength corresponding to the periodic solution of the variational equation defines an additional stability transition curve which has only been observed previously in the relaxation limit. We show that this transition curve, which provides a consistent connection between the sinusoidal and relaxation limits, is O(ε2) and could not have been delected in O(ε) analyses. We determine the analytical expression for this stability transition curve to O(ε31) and show very favorable agreement with numerical results we obtained using an Adams-Gear method.