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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 967395, 10 pages
http://dx.doi.org/10.1155/2014/967395
Research Article

Bifurcation of Safe Basins and Chaos in Nonlinear Vibroimpact Oscillator under Harmonic and Bounded Noise Excitations

1Department of Mathematics, Foshan University, Foshan 528000, China
2Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China

Received 10 August 2014; Revised 5 October 2014; Accepted 7 October 2014; Published 21 December 2014

Academic Editor: Qingdu Li

Copyright © 2014 Rong Haiwu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The erosion of the safe basins and chaotic motions of a nonlinear vibroimpact oscillator under both harmonic and bounded random noise is studied. Using the Melnikov method, the system’s Melnikov integral is computed and the parametric threshold for chaotic motions is obtained. Using the Monte-Carlo and Runge-Kutta methods, the erosion of the safe basins is also discussed. The sudden change in the character of the stochastic safe basins when the bifurcation parameter of the system passes through a critical value may be defined as an alternative stochastic bifurcation. It is founded that random noise may destroy the integrity of the safe basins, bring forward the occurrence of the stochastic bifurcation, and make the parametric threshold for motions vary in a larger region, hence making the system become more unsafely and chaotic motions may occur more easily.