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The two main objectives of this article are (1) to determine the bifurcating periodic responses accurately and to study the effects of time delay and feedback gain on the steady state response in autonomous oscillators by means of the residue harmonic balance method; and (2) to study the dynamics of both the autonomous and non-autonomous Duffing-van der Pol oscillators with delayed position feedback as examples. The results of the steady state response to frequency and amplitude excitation versus the time delay, positive and negative feedback gains are studied. Both saddle-node and subcritical Hopf bifurcations are found and discussed. The obtained second-order solutions are in excellent agreement with numerical integration results. By combining both harmonic balance and polynomial homotopy continuation techniques, the superharmonic resonance responses of the non-autonomous oscillator are investigated in order to obtain all possible steady state solutions. The effects of the feedback gains and time delay on the steady state response are found analytically. It is shown that the fundamental and superharmonic amplitudes decrease with increasing feedback gain. Furthermore, the multiple symmetric solutions of the response amplitudes versus those of external excitation are detected when time delay becomes smaller. [ABSTRACT FROM AUTHOR]

Abstract: When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time. [Copyright &y& Elsevier]

Abstract: The residue harmonic balance is developed for coupled systems exemplified by the damped Duffing resonator driven by a van der Pol oscillator. This technique combines the features of harmonic balance and parameter bookkeeping to obtain approximate solutions to any desired accuracy. For the two degrees of freedom system, the zeroth-order approximation uses just one Fourier term resulting in a set of two nonlinear algebraic equations that can be solved analytically. The unbalanced residue due to Fourier truncation is improved by solving linear equations that automatically increase the number of terms to balance the previous truncation completely. Highly accurate bifurcation frequencies for various parameters are obtained and are verified by numerical integration results. The effects of the coupling stiffness on the angular frequency and amplitude of steady state response are investigated. The effects of the nonlinear damping are also studied. [Copyright &y& Elsevier]

Abstract: In this paper, a powerfully analytical technique is proposed for predicting and generating the steady state solution of the fractional differential system based on the method of harmonic balance. The zeroth-order approximation using just one Fourier term is applied to predict the parametric function for the boundary between oscillatory and non-oscillatory regions of the fractional van der Pol oscillator. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy of the solutions successively. The highly accurate solutions to the angular frequency and limit cycle of fractional van der Pol oscillator are obtained and compared. The results reveal that the technique described in this paper is very effective and simple for obtaining asymptotic solution of nonlinear system having fractional order derivative. [Copyright &y& Elsevier]