Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay

In this paper, a Duffing-van der Pol oscillator having fractional derivatives and time delays is investigated by the residue harmonic method. The angular frequencies and limit cycles of periodic motions are expanded into a power series of an order-tracking parameter and the unbalanced residues resulting from the truncated Fourier series are considered iteratively to improve the accuracy. The periodic bifurcations are examined using the fractional order, feedback gain and time delay as continuation parameters. It is shown that jumps and hysteresis phenomena can be delayed or removed. Transition from discontinuous bifurcation to continuous bifurcation is observed. The approximations are verified by numerical integration. We find that the proposed method can easily be programmed and can predict accurate periodic approximations while the system parameters being unfolded.