Residue harmonic balance approach to limit cycles of non-linear jerk equations

A residue harmonic balance is established for accurately determining limit cycles to parity- and time-reversal invariant general non-linear jerk equations with cubic non-linearities. The new technique incorporates the salient features of both methods of harmonic balance and parameter bookkeeping to minimize the total residue. The residue is separated into two parts in each step; one conforms to the present order of approximation and the remaining part for use in the next order. The corrections are governed by a set of linear ordinary differential equations that can be solved easily. Three specific cases of non-linear jerk equations are given to illustrate the validity and efficiency. The approximations to the angular frequency and the limit cycle are obtained and compared. The results show that the approximations obtained are in excellent agreement with the exact solutions for a wide range of initial velocities. The new technique is simple in principle and can be applied to other non-linear oscillating systems.