A pair of van der Pol oscillators coupled by fractional derivatives

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators: 1[graphic not available: see fulltext]2[graphic not available: see fulltext] where Dαx is the order α derivative of x(t), and 0<α<1. We use a two-variable perturbation method on the system’s corresponding variational equations to derive expressions for the transition curves separating regions of stability from instability in the α, γ parameter plane. The perturbation results are validated with numerics and through direct comparison with known results in the limiting cases of α=0 and α=1, where the fractional coupling reduces to position coupling and velocity coupling, respectively.