After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE: x″+(δ+εcost)x+cDαx=0 where Dαx is the order α derivative of x(t), where 0 < α < 1. We use the method of harmonic balance and obtain both a lowest order approximation as well as a higher order approximation for the n = 1 transition curves. We also obtain an expression for the n = 0 transition curves.

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A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed = 0 system. The resulting amplitude-delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration.

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