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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.

Nonsmooth factors arise naturally in engineering applications, such as impacts, collisions, and dry frictions [

It is well known that, by calculating the distance between the stable and unstable manifold, Melnikov’s method [

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

Consider a classical Duffing vibroimpact oscillator with bilateral constrains under both harmonic and bounded random noise excitations governed by the following equation:

The physical model of (

When

Equation (

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 (

Now we give some numerical results to verify the analytic conditions given by (

Chaotic area of system (

Three different simulation points in Figure

Numerical results of (

Time history of

Phase plot

Numerical results of (

Time history of

Phase plot

The response time history of system (

Figures

Numerical results of (

Time history of

Phase plot

Numerical results of (

Time history of

Phase plot

The response time history of system (

Figures

It is well known, from the theory of nonlinear oscillation, that if an oscillator with hardening nonlinear stiffness is subjected to sinusoidal excitation, the response may exhibit sharp jumps in amplitude. This jump behavior is associated with the fact that, over a range of the values of the ratio of excitation frequency to the natural frequency of the degenerated linear oscillator, the response amplitude is triple-valued. Therefore the system should have two stationary responses which depend on the initial condition. However, it is a disputable problem whether there are more than one stationary response if an oscillator with hardening nonlinear stiffness is subjected to random excitations [

The third simulation point is

Numerical results of (

Time history of

phase plot

Alternative to the Melnikov function and Lyapunov exponent method, there is another method to identify the rising of chaos. One of these is to determine the global structures of the system and one of these global structures is the boundary of safe basin. The safe basin boundaries of attractors are usually fractal and naturally incursive since the coexistence of period and chaotic attractors. They are related to homoclinic or heteroclinic intersections of stable and unstable manifolds of the saddle points in the system and chaos often arises in such system. The decrease of the safe basin’s area is called basin erosion and will be discussed in this section.

In some time the limitation of the vibration amplitude may be more important, since the structure of the system will be destroyed when the amplitude of the vibration passes through a critical value and thus leads to the researches of the safe basins [

In this paper, the evolution of the safe basins of system (

Erosion of safe basins in system (

Erosion of safe basins in system (

The safe basins shown in Figure

When

Next, we consider the effect of the random noise on the safe basins, the parameters in system (

Figures

When

Overall, random noise may destroy the integrity of the safe basins boundary, bring forward the occurrence of the stochastic bifurcation, and hence make the system become more unsafely. The threshold value of the stochastic dynamical system form condition (

The erosion of the safe basins and chaotic motions of a nonlinear vibroimpact oscillator under harmonic and bounded random noise is studied. Melnikov’s method in the deterministic vibroimpact system is extended to the analysis of homoclinic bifurcations and chaos in the stochastic case. The results reveal that the threshold amplitude

Although the theory of stochastic bifurcation has been advanced to a new level in the last decade, there remain a lot of problems to be solved. Even the definition of stochastic bifurcation needs to be improved. In this paper, we suggest an alternative definition for stochastic bifurcation based on the analysis of the safe basins of a softening Duffing oscillator subject to deterministic harmonic and bounded random excitations, which focuses on a sudden change in the character of the safe basins of the dynamical system as the bifurcation parameter passes through a critical value. This definition applies equally well either to the stochastic bifurcation or to the deterministic bifurcation. However, the application of the definition for real systems needs more effect. The analysis shows that the random noise causes the two bifurcation points

In the paper, Melnikov’s methods and bifurcation of safe basins are the main research methods. In the fact, there are other effective method to verify the chaos, for instance, topological horseshoes method which has been successfully applied in many works [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work reported in this paper was supported by the National Natural Science Foundation of China under Grant no. 11401096 and 11326123, the Natural Science Foundation of Guangdong Province under Grant no. S2013010014485 and S201310012463, and Special fund of the Guangdong College discipline construction under Grant nos. 2013KJCX0189 and 2013B020314020.