Extension and improvement to the Krylov-Bogoliubov methods using elliptic functions

The Elliptic Multiple Scales Method for a Class of Autonomous Strongly Non-Linear Oscillators

“…It provides the second approximate term x to correct the approximation x , in agreement with the LP method [14]. It gives also the modulation equations of amplitude and phase, in agreement with the elliptic KBM method [10]. Finally, it o!ers the possibility for deriving higher order approximations in an interactive way.…”

“…For instance, Margallo et al [8,9] presented an elliptic HB method using generalized Fourier series and elliptic functions. Yuste and Bejarano [10] developed an elliptic KBM method and Coppola and Rand [11] used symbolic computation to implement an averaging method with elliptic functions. All the methods mentioned above have their own advantages to obtain approximate analytical solutions.…”

Periodic Solutions of Strongly Non-Linear Oscillators by the Multiple Scales Method

“…In particular, the use of Jacobian elliptic functions gives an excellent approximation of the periodic orbits, even near the separatrices in the self-excited oscillators (2), just prior to a homoclinic saddle-loop connection. For instance, Barkham and Soudack [8], and Yuste and Bejarano [9] used the (KBM) method to provide approximate solutions of a strongly non-linear oscillator in terms of Jacobian elliptic functions. Also, Margallo et al [10,11] presented an elliptic HB method using generalized Fourier series and elliptic functions.…”

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

“…For small ), and 9(x) is linear, the classical perturbation method [1] can be applied to the problem of determining limit cycles approximately. Generalizations have been obtained for the cases where g(x) is linear plus cubic polynomial terms [2][3][4][5] and where 9 (x) is an arbitrary function [6][7][8][9][10]. For moderately large A and 9(x) is an arbitrary nonlinear function, there are the two timescale harmonic balance method [11] and the perturbation-iterative method [12].…”