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Bifurcation of the periodic motion in nonlinear delayed oscillators

We use the residue harmonic balance scheme to study the periodic motions of a class of second-order delay-differential equations with cubic nonlinearities near and after Hopf bifurcation. The multiple solutions are found by homotopy continuation. Then, the approximation to any desired accuracy for a specific solution is captured by solving linear equations iteratively. The second-order solutions give good predictions for the frequency and amplitude, which are verified by the Runge–Kutta numerical solutions. Two typical examples, the temporal dynamics of the delay Liénard oscillator and the delay feedback Duffing system, are studied and compared. The results show how to trace analytically the relevant effect of the stiffness coefficient and the time delay on the dynamics and on the number of periodic solutions, even for large values of the bifurcation parameters.