Limit cycle analysis of a class of strongly nonlinear oscillation equations

Abstract

The limit cycle of a class of strongly nonlinear oscillation equations of the form/2 + g(u) = ef (u, iz) is investigated by means of a modified version of the KBM method, where e is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to e = 0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large e.

Elliptic harmonic balance method for two degree-of-freedom self-excited oscillators

“…Davis [1] used the elliptic functions to study non-linear oscillators in 1960s. After that, some new analytical methods were proposed by combining elliptic functions with classical techniques for non-linear oscillators, e.g., the elliptic KB method [2], the elliptic KBM method [3], the elliptic perturbation method [4], and the elliptic Lindstedt-Poincaré method [5]. They have been found applications in various strongly non-linear problems, e.g., the elliptic Lindstedt-Poincaré method was used to study the homoclinic connections of strongly selfexcited non-linear oscillators [6], the elliptic perturbation method was implemented to analyze the generalized Rayliegh oscillator equation [7], and the elliptic KB method was used to analyze a two DOFs oscillating system [8].…”

On the Shape of Limit Cycles That Bifurcate from Isochronous Center

“…The study of limit cycles of real general planar vector field is closely related to Hilbert's 16th Problem. As to the strongly nonlinear oscillation equation dx / dt = y , dy / dt = g ( x ) + ɛ f ( x , y ), in [ 2 ], the first two order approximate expressions of limit cycles for small positive parameter ɛ were studied by the generalized KBM method, and, in [ 3 ], the shape of the limit cycles for moderately large positive parameter ɛ was plotted by using the perturbation-incremental method.…”

A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems

“…Until now, various improved and novel perturbation techniques have been developed such as the modified L-P method [3], the elliptic L-P method [4], the elliptic perturbation method [5], the nonlinear time transformation method [6], the generalized averaging method [7], the nonlinear scales method [8], the modified KBM method [9], etc. In particular, Chen et al [10] pointed out that the use of conditions of constant phase angles in the perturbation procedures could provide more accurate results for limit cycle analysis, especially for the strongly nonlinear cases.…”