Impact of coupling on the road to synchronization of two coupled Van der Pol oscillators

Abstract

Synchronized dynamics of two coupled van der Pol oscillators x and y balances between the attracting forces and the prevention of synchrony represented by the coupling μ and the half difference of their natural frequencies Δω. The current paper investigates two regimes of such balance: (i) μ slightly exceeds Δω, and (ii) μ is large. In case of regime (i), the oscillators are shown to quickly attain the neighborhood of the limit cycle of a complex geometry; their pre-limit behavior can be characterized by the vertical mismatch x−y. If μ→∞, case (ii), the shape of the solutions of the underlying differential equations promptly follows that of the limit cycle, which is the solution of a single van der Pol equation, but the tendency to the limit cycle is slow, being proportional to 1/μ. Further, the mismatch between the oscillators disappears as μ increases. During the transition between the regimes occurred with the growth of μ, the vertical mismatch and the time required to solutions to reach the neighborhood of the limit cycle exhibit the Λ- and V-shapes. These shapes attain the maximum and, respectively, minimum at moderate values of μ that were earlier suggested by studies of synchronization between solar activity components.