Forward residue harmonic balance for autonomous and non-autonomous systems with fractional derivative damping

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.