The Residue Harmonic Balance in Bifurcation, Limit Cycle and Wave Analyses

The harmonic balance method replaces ODEs in time to algebraic equations in frequency describing frequency distribution in the steady state response of the nonlinear system. The residue harmonic balance concentrates on the annihilation of the residues and can double the effective harmonic terms in each iteration step and can be considered as an extension of the Newton iteration from algebraic equations to ODEs. The systems being considered include rational and irrational nonlinearity; integral and fractional derivatives; and single and multiple degrees of freedom. The method is extended to steady state response of time delay systems and of wave described by PDEs. Examples include famous oscillators with fractional derivative damping and with time delay; applications in aerospace and in structural dynamics.