On perturbation procedure for limit cycle analysis

Abstract

In the use of perturbation procedure for analysing the limit cycle of an autonomous system, people usually assumed an initial condition ẋ(0) = 0. However, this initial condition is often simplified to x ̇nh(0) = 0, (n= 0, 1, 2,...) and taken as an additional condition to determine a constant, say bn, of the homogeneous solution xn = an cosτ + bn sin τ of the perturbation equation of each order. Nevertheless, this commonly accepted procedure will lead to large errors, especially for the case of larger values of the parameter ε. In this paper, a condition of constant phase angle Øn in the homogeneous solutions xnh = Ancos (τ + Øn) (n = 0, 1, 2,...) is adopted, i.e. Ø1 = Ø2 = ... = Ø. The limit cycles obtained by use of the present condition and corresponding perturbation procedure are in good agreement with the numerical results of the Runge-Kutta integration, and with the analytical expression obtained by the incremental harmonic balance method even in the case of the parameter ε = 1. © 1990.