Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

Abstract An improved version of a Krylov-Bogoliubov method that gives the approximate solution of the non-linear cubic oscillator x + c 1 x + c 3 x 3 + ef(x, x dot ) = 0 in terms of Jacobi elliptic functions is described. Compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual Krylov-Bogoliubov method (which gives the approximate solution in terms of circular functions). These expressions are especially simple for quasilinear (c3 = 0) and quasi-pure-cubic (c1 = 0) oscillators. Two types of cubic oscillators have been used as examples: the linear damped oscillator f(x, x dot ) = x dot , and the van der Pol oscillator f(x, x dot ) = (α − βx 2 ) x dot . The approximate solutions of these quasilinear and quasi-pure-cubic oscillators are simple and accurate. The influence of the non-linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied.

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