Stability of in-phase and out-of-phase synchronization modes in Van der Pol-Duffing oscillators coupled by fractional order derivative

In this paper we consider a pair of Van der Pol-Duffing oscillators coupled by fractional order time derivative for in-phase and out-of-phase synchronization. The stability regions in parametric space defined by coupling strength and fractional order are determined. The transition curves separating stable and unstable regions are derived by applying perturbation method. The effects of Duffing nonlinearity on the synchronization characteristics are analyzed, and the regions of bi-stability are explored.