Abstract
The present paper focuses on the noise-induced chaos in aΦ6oscillator with nonlinear damping. Based on the stochastic Melnikov approach, simple zero points of the stochastic Melnikov integral theoretically mean the necessary condition causing noise-induced chaotic responses in the system. To quantify the noise-induced chaos, the Poincare maps and fractal basin boundaries are constructed to show how the system's motions change from a periodic way to chaos or from random motions to random chaos as the amplitude of the noise increases. Three cases are considered in simulating the system; that is, the system is excited only by the harmonic excitation, by both the harmonic and the Gaussian white noise excitations, or by both the bounded noise and the Gaussian white noise excitations. The results show that chaotic attractor is diffused by the noises. The larger the noise intensity is, the more diffused attractor it results in. And the boundary of the safe basin can also be fractal if the system is excited by the noises. The erosion of the safe basin can be aggravated when the frequency disturbing parameter of the bounded noise or the amplitude of the Gaussian white noise excitation is increased.
Highlights
In recent years, stochastic Melnikov method has been applied to study the effects of noise on homoclinic or heteroclinic bifurcation and noise-induced chaos [1,2,3,4,5,6]
The present paper demonstrates the effects of the bounded noise and Gaussian white noise on Poincare maps and the boundary of the safe basin, respectively, in the triplewell Φ6 potential nonlinear oscillator
The results show that the Homoclinic bifurcation condition is satisfied, the response of the system undergoes harmonic motions rather than chaotic motions
Summary
Stochastic Melnikov method has been applied to study the effects of noise on homoclinic or heteroclinic bifurcation and noise-induced chaos [1,2,3,4,5,6]. Zhu et al [7,8,9] and Xu et al [10,11,12,13] used the stochastic Melnikov method and the Poincare maps to show the random chaos of the systems, but in spite of the fact that the chaotic system is sensitive to the initial values, they did not concern the safe basin of the systems to find how the initial conditions influence the system’s motion. Gan [14, 15] studied the erosion of the boundaries of the safe basins of a one well system and a double wells system after applying the stochastic Melnikov method. It is necessary for us to seek for better understanding of the responses of higher order term nonlinear systems with both white noise and bounded noise excitations. It can be shown that the functions of the noise are continuous and bounded, which are required in the derivation of Melnikov function
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