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This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems. In this solution procedure, no small parameter is assumed. The harmonic residue of balance equation is separated in two parts at each step. The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement. The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again. Three kinds of different differential equations involving general, fractional and delay ordinary differential systems are given as numerical examples respectively. Highly accurate limited cycle frequency and amplitude are captured. The results match well with the exact solutions or numerical solutions for a wide range of control parameters. Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems. Moreover, the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.

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]

Abstract: This paper investigates the steady state bifurcation of a periodically excited system subject to time-delayed feedback controls by the combined method of residue harmonic balance and polynomial homotopy continuation. Three kinds of delayed feedback controls are considered to examine the effects of different delayed feedback controls and delay time on the steady state response. By means of polynomial homotopy continuation, all the possible steady state solutions corresponding the third-order superharmonic and second-subharmonic responses are derived analytically, i.e. without numerical integration. It is found that the delayed feedback changes the bifurcating curves qualitatively and possibly eliminates the saddle-node bifurcation during resonant. The delayed position-velocity coupling and the delayed velocity feedback controls can destabilize the steady state responses. Coexisting periodic solutions, period-doubling bifurcation and even chaos are found in these control systems. The neighborhood of the periodic solutions is verified numerically in the phase portraits. The various effects of time delay on the steady state response are investigated. Many new phenomena are observed. [Copyright &y& Elsevier]

Abstract: Both the autonomous and non-autonomous systems with fractional derivative damping are investigated by the harmonic balance method in which the residue resulting from the truncated Fourier series is reduced iteratively. The first approximation using a few Fourier terms is obtained by solving a set of nonlinear algebraic equations. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. Multiple solutions, representing the occurrences of jump phenomena, supercritical pitchfork bifurcation and symmetry breaking phenomena are predicted analytically. The interactions of the excitation frequency, the fractional order, amplitude, phase angle and the frequency amplitude response are examined. The forward residue harmonic balance method is presented to obtain the analytical approximations to the angular frequency and limit cycle for fractional order van der Pol oscillator. Numerical results reveal that the method is very effective for obtaining approximate solutions of nonlinear systems having fractional order derivatives. [ABSTRACT FROM AUTHOR]