The application of the undetermined fundamental frequency for analyzing the critical value of chaos

Abstract

The simple approach to improve the computational precision of Melnikov method is presented by using the undetermined fundamental frequency and normal form method. We construct the improved Melnikov expression for a triple-well nonlinear oscillator subject to principal parametric resonance and external excitation. For the occurrence of chaos, the approxime threshold values of chaotic motion are obtained from the Homoclinicity and Heteroclinicity points of view. It depends on the introduction of undetermined fundamental frequency, and adopting new time transformation for fulfilling the homoclinic and heteroclinic orbits, so that the effect of disturbing parameter can be easily detected and embodied in the Melnikov operation. As is illustrated, the explicit applications show that the improved results coincide very well with the results of numerical simulation.