The limit cycles of the generalized Rayleigh-Liénard oscillator

Abstract

The limit cycles of the generalized Rayleigh and mixed (Rayleigh-Liénard) oscillator X ̈ + AX + 2BX 3 + ε(z 3 + z 2X 2 + z 1X 4 + z 4 dot X 2) dot X = 0 , for A and B > 0, are studied by using the Jacobian elliptic functions with the generalized harmonic balance method. The transitory motion, and consequently the limit cycles and their stability, are also studied quantitatively with a generalized approximation of the Krylov-Bogoliubov slowly varying amplitude and phase type, giving the radius, frequency, and energy of the limit cycles. Whether there are zero, one or two limit cycles, and their stability, depends on the values of the parameters z i , i = 1, 2, 3, 4. These solutions are interesting because they do not depend on the value of ε. For the simple cases of only two non-zero z i parameters, plots of the universal functions are given as well as the limit cycle radii. Approximate solutions in the general case are found to coincide with certain exact solutions of the mixed-type oscillators.