Journal of Applied Mathematics

Volume 2014 (2014), Article ID 967395, 10 pages

http://dx.doi.org/10.1155/2014/967395

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

^{1}Department of Mathematics, Foshan University, Foshan 528000, China^{2}Department 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.

#### 1. Introduction

Nonsmooth factors arise naturally in engineering applications, such as impacts, collisions, and dry frictions [1]. A considerable amount of research activities have focused on nonsmooth dynamical systems, including vibroimpact systems, collisions dynamics, chattering dynamics, and stick-slip motions, in recent years. In practice, engineering structures are often subjected to time dependent loadings of stochastic nature, such as the natural phenomena due to wind gusts, earthquakes, ocean waves, and random disturbance or noise which always exists in a physical system. The influence of random excitation on the dynamical behavior of a vibroimpact dynamical system has caught the attention of many researchers. Many effective methods have been developed, for example, linearization method used by Metrikyn [2], quasistatic approach method used by Stratonovich [3], exponential polynomial fitting method proposed by Zhu [4], Markov process method used by Jing and Young [5], stochastic averaging method used by Xu et al. [6, 7], variable transformation method used by Zhuravlev [8], energy balance method used by Iourtchenko and Dimentberg [9], mean impact Poincaré map method used by Feng and He [10], path integration method used by Iourtchenko and Song [11], and numerical simulation method used by Dimentberg et al. [12]. In [13], the authors tried to review and summarize the existing methods, results, and literature available for solving problems of stochastic vibroimpact systems. However, most researches focused on responses statistics, such as statistic moment and probability density function of the vibroimpact oscillator, and few are focused on the bifurcations and chaos of the stochastic vibroimpact dynamical systems.

It is well known that, by calculating the distance between the stable and unstable manifold, Melnikov’s method [14] gives a powerful approximate tool for investigating chaotic motions in deterministic smooth system. However, classical Melnikov’s method is not directly appropriate in the nonsmooth system. Some effective Melnikov’s methods have been proposed for deterministic piecewise smooth dynamical systems [15, 16] and nonlinear vibroimpact dynamical systems [17]. To our knowledge, few Melnikov’s methods are well developed for the stochastic vibroimpact dynamical systems.

In this paper, the bifurcation of safe basins and chaos of a nonlinear vibroimpact oscillator under both harmonic and bounded noise excitation are investigated. The impact considered here is an instantaneous impact with restitution factor . The paper is organized as follows. In Section 2, Melnikov’s method is extended to the analysis of homoclinic bifurcation in the stochastic vibroimpact system, and the conditions for the onset of chaos are derived in the mean square value sense. In Section 3, the variations of safe basins are presented numerically when one changes the amplitude of the harmonic excitation both in the deterministic and stochastic cases. Conclusions are presented in Section 4.

#### 2. The Stochastic Melnikov Function

##### 2.1. Theoretical Analysis

Consider a classical Duffing vibroimpact oscillator with bilateral constrains under both harmonic and bounded random noise excitations governed by the following equation: where the dot indicates differentiation with respect to time , is a small scale, , represents the intensity of the nonlinear term, is the damping coefficient, and are amplitude and frequency of the deterministic excitation, respectively, and is the restitution factor to be a known parameter of impact losses, , whereas subscripts “minus” and “plus” refer to value of response velocity just before and after the instantaneous impact. Thus and are actually rebound and impact velocities of the mass, respectively. They have the same magnitude whenever ; therefore this special case is that of elastic impacts, whereas in case some impact losses are observed. denotes the intensity of the random excitation and is a bounded noise process, which was introduced by Stratonovich [18]: where is the frequency of the random excitation, is a standard Wiener process, and a uniformly distributed random number in . is the random disorder which describes random temporal deviations of the excitation frequency from its expectation or mean . The process has the following power spectral density [19]: Obviously is a bounded random process. Periodic-in-time excitation with a “random disorder,” or random phase modulation appears in structural dynamics with traveling loadings and/or structures, having certain imperfect spatial periodicity in certain problems of aeroelasticity [19]. This process will be assumed to be narrow-band, which is clearly seen to be in the case provided that . We assume that in this paper. The bounded noise model (2) is a suitable model and many researches have been done on the responses of nonlinear system under bounded noise excitation [20, 21]. Obviously, bounded noise has continuous sample function; hence Melnikov’s method may be used in the stochastic system (1).

The physical model of (1) can be viewed as the motion of a mass with harden stiffness under both harmonic and bounded random noise external excitations, while two collision obstacles are placed before and after the equilibrium position with distance 1.

When , system (1) reduces to an unperturbed deterministic vibroimpact system. Using stability analysis, one collects two center-type fixed points and , and a saddle-type fixed point . Denotes ; two homoclinic orbits connecting the two center-type fixed points and saddle-type fixed point are where .

Equation (4) is exactly the solution of the unperturbed deterministic system (1) as without vibroimpact, and similar system has been discussed in [17] by using the Melnikov method. Using the results in [17, 22], one obtains the stochastic Melnikov function for homoclinic orbits of system (1): where , and are the deterministic parts of the random process ; they are caused by the deterministic harmonic excitation, and represents the stochastic term which is caused by the bounded noise :

According to the dynamic theory, the stable and the unstable manifolds will intersect transversely with each other which means chaos will occur when there exist simple zeros in Melnikov function (5) in the deterministic case . However, in the stochastic case , the Melnikov function measures the random distance between the stable and the unstable manifolds. In this case, the threshold value for the rising of the chaotic motion depends on the property of the random excitation process and may deviate from the one for the deterministic case. In order to analyze the simple zero points of in the statistics sense, one considers the following equations: where represents the mathematics expectation. Since , the condition for the onset of chaotic motion in the mean square value sense is where Then, from (8), the condition of the occurrence of chaotic motions of system (1) is

##### 2.2. Numerical Results

Now we give some numerical results to verify the analytic conditions given by (8) and (10). The parameters of system (1) are given by The variation of the threshold value versus the bounded noise amplitude is plotted in Figure 1 for the onset of chaotic monition in system (1), the solid line represents the analytic results given by (10), and the dashed line represents the numerical simulations according the criterion of the largest Lyapunov exponent using the algorithm presented by Wolf et al. [23]. Usually, calculating the Lyapunov exponents is regarded as the simplest method to verify the existence of chaotic behavior. However, the Melnikov method and the erosion of the safe basins are mainly discussed in this paper, while the largest Lyapunov exponent is only used to verify the efficiency of the above two methods. The largest Lyapunov exponent of system (1) is positive in the area above the dashed line, which means the occurrence of the chaotic motion, while the largest Lyapunov exponent of system (1) is negative in the area below the dashed line, which means no occurrence of the chaotic motion. It can be seen clearly from Figure 1 that Melnikov’s condition (10) is only a necessary condition for the occurrence of chaotic motion in the Lyapunov sense. It can also be seen that the threshold value will decrease when increase thus make the chaotic motions occur more easily.