Generalized Fourier series and limit cycles of generalized van der Pol oscillators
Abstract
A very simple generalization, using Jacobian elliptic functions, of the usual Fourier series, appropriate for nonlinear systems, is used to study the first order approximate solution of the generalized van der Pol oscillators Ẍ+AX+2BX ^{3}+∊(z _{3}+z _{2}X ^{2}+z _{1}X ^{4}) dotX=0 . A generalized harmonic balance method is used to determine the limit cycles. The cases A > 0, B > 0 and A < 0, B > 0 are considered in some detail. For given values of the parameters z_{i}, the values of A and B for which limit cycles exist are found as functions of m. Numerical values for the radius, frequency, and energy of the limit cycles are given. The presence of zero, one, two, three or four limit cycles depends on the value of the parameters of the equation. Stability, bifurcations of fixed points, and the limit cycle stability are studied qualitatively.
 Publication:

Journal of Sound Vibration
 Pub Date:
 February 1990
 DOI:
 10.1016/0022460X(90)90456A
 Bibcode:
 1990JSV...136..453G