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Highlights: [•] The residue harmonic balance is again introduced in bifurcation analysis. [•] The periodic bifurcations are examined using the fractional order, feedback gain and time delay as continuation parameters. [•] All possible steady state solutions are derived analytically without numerical integration. [•] The fractional order, feedback gain and time delay change the bifurcating curves qualitatively. [•] Control strategies are proposed to avoid jumps and hysteresis phenomena using trials and errors. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the residue harmonic balance method to generate periodic solutions for nonlinear evolution equations. A PDE is firstly transformed into an associated ODE by a wave transformation. The higher-order approximations to the angular frequency and periodic solution of the ODE are obtained analytically. To improve the accuracy of approximate solutions, the unbalanced residues appearing in harmonic balance procedure are iteratively considered by introducing an order parameter to keep track of the various orders of approximations and by solving linear equations. Finally, the periodic solutions of PDEs result. The proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the sophisticated implicit functions as appearing in most methods. [Copyright &y& Elsevier]

Highlights: [•] More general forms of damping in terms of fractional derivative and time delay are investigated. [•] The damping characteristics in Duffing–van der Pol oscillators are taken as examples. [•] The steady state limit cycle bifurcations are solved by the residue harmonic balance. [•] Rapid convergence is obtained by solving linear ODE. [•] Qualitative differences in the damping characteristics for these dampers are found for the first time. [ABSTRACT FROM AUTHOR]