1 Introduction

The response analysis of stochastic nonlinear systems has been extensively studied in the past several decades,and many methods^{[1,2,3,4,5,6,7,8]} have been developed for response probability density functions (PDFs). However,most of these methods are only effective to get stationary PDFs. Therefore,the studies of nonstationary or transient PDFs^[9,10] and evolutionary stochastic responses^[11] become more and more attractive. Stochastic bifurcation^{[12,13,14,15]} is an important research topic for the response of nonlinear systems. In a phenomenological sense,it can be defined as the qualitative change in the shape of the stationary PDF^[16] . Zhu et al.^[12] pointed out that the peaks in stationary PDFs represented the stable motion of noise-free systems,and stochastic P-bifurcation occurred when the shape of the stationary PDF changed from bimodal to unimodal or vice versa. Therefore,it is necessary to analyze the stochastic responses in combination with the global properties of the noise-free system.

The generalized cell mapping (GCM) method was firstly proposed by Hsu^[17] . Since then, many relevant modified methods have been developed^[18,19,20] ,and are widely used to analyze the global properties and global bifurcations of nonlinear systems^[21,22,23] . They are effective tools for the stochastic response analyses of systems with noise. Sun and Hsu^[24] and Sun^[25] successfully examined the first-passage time probability and the random vibration analysis of some nonlinear systems by the GCM method. Wu and Zhu^[26] investigated the stochastic analysis of a pulsetype prey-predator model. Yue et al.^[27,28] studied the transient and steady-state responses of oscillators with different noise excitations.

There exist lots of mechanical and engineering models,in which the force-deflection relationships include real-power exponents^[29,30] . These viscoelastic systems can reveal material properties more properly. This kind of systems without noise have been extensively studied in recent years^[31,32] . Considering the stochastic excitation,Liu et al.^[33] investigated the noiseinduced chaos in the oscillator with real-power exponents of damping and restoring force by the random Melnikov method. However,very few attention has been paid to the stochastic responses of these systems because of the existence of real-power restoring force.

In this paper,the stochastic response analysis of a system with real-power restoring force is studied by the GCM method. The rest of this paper is arranged as follows. In Section 2, the GCM method for the stochastic response analysis is reviewed briefly. Then,the model of the system and its global properties are presented in Section 3. In Section 4,we focus on the stochastic response analysis of the system,including transient PDFs,stationary PDFs,and stochastic P-bifurcation. Finally,some conclusions are drawn in Section 5. 2 GCM method for stochastic response analysis

We consider an N-dimensional dynamical system under stochastic excitation. When the GCM method is used for its stochastic response analysis,the continuous Markov vector process of the system is reduced to a Markov chain by discretization of both the time axis and the state space^[24,25,26] .

Let x(t) denote the response process of the state variables,∆t denote the length of the discrete time intervals,and p(x₀,t_n) denote the PDF at t_n = n∆t,n = 0,1,2,· · · . Then,the PDF at the subsequent instant t_n+1 can be computed by

where q(x,∆t |x₀,0 ) represents the homogeneous transition PDF of x(t) = x at t = ∆t with the condition x(t) = x₀ at t = 0. An interest bounded region in R^N is chosen and divided into N_c cells of the same size. Each cell is indexed by an integer ranging from 1 to N_c. Because point states are replaced by cell states,we have where i and j are the cells which contain the states x₀ and x,respectively. C_i(j) is the domain occupied by the cell i(j). If p_i⁽ⁿ⁾ is used as the probability that the response is located in the cell i at t_n ,then we can get the evolutionary equation as follows:

Assume that Q is the one-step transition probability matrix formed by the elements q_ji (i,j = 1,2,· · · ,Nc),which are computed numerically by the same method used in Ref. ^[24],and P⁽ⁿ⁾ = (p₁⁽ⁿ⁾,p₂⁽ⁿ⁾ 2,· · · ,p_Nc⁽ⁿ⁾) is the probability distribution vector at t_n. Then,the idea of Eq. (3) can be rewritten as

The transient response is time-variation,and depends on the initial probability distribution P⁽⁰⁾ . Different initial distributions result in different evolution transient response processes. However,the stationary response is independent of t. It reflects the behavior of the system after a very long time. Therefore,the transient probability distribution P⁽ⁿ⁾ at t_n can be obtained by assigning an initial probability distribution P⁽⁰⁾ and using Eq. (4),i.e.,

As n → ∞,the transient probability distribution approximates a stationary one. When the probability distribution P⁽ⁿ⁾ is obtained,we can compute the corresponding PDF of the transient or stationary response. 3 Model and its global properties

In this section,a system with the non-negative real-power restoring force under harmonic and Gaussian white noise excitations is considered. It also owns a quadratic damping,which is one of the basic damping mechanisms. The non-dimensional dynamical equation is

where x is the state variable of the displacement. The dot denotes derivative with respect to the time t. ζ ,m,and β are all positive parameters of the system. α is an arbitrary positive real number,which is bigger than 1. f and ω are the amplitude and the frequency of the harmonic excitation,respectively. W(t) represents a unit Gaussian white noise with the correlation function

E(W(t) W(t + τ)) = δ(τ).

D is treated as the intensity of the stochastic noise excitation. For convenience,the system (6) can be rewritten as where

For the system (7) without noise (D= 0),the GCM method for the global analysis of the deterministic systems in Ref. ^[19] is used in

Then,we can obtain the global properties on the Poincar´e section (see Fig. 1). There are two coexistent period-1 attractors in the phase space,marked by A1 and A2,respectively. The corresponding attraction basins are denoted as B1 and B2,respectively,and B(1,2) is the basin boundary between them. In addition,S(1,2) is a period saddle located at the boundary B(1,2). SM and UM represent the stable manifold and the unstable manifold,respectively. The two attractors are stable motion of the system response,while the saddle corresponds to the unstable transient-state. When the stochastic analysis of the system with noise is carried out,the number of the attractors is related to the shape of the stationary PDF. Moreover,the saddle and its invariant manifolds,especially the unstable one,play very important roles in the evolution of the PDFs.

4 Response analysis of system under Gaussian white noise excitation

To analyze the stochastic response of the system under random excitation with the GCM method,we reset the interest region Ω to be {−3.2 ≤ x ≤ 2.2 ,−1.5 ≤ y ≤ 1.5},and design a partition of 100 × 100 cells in it. 10 × 10 interior representative points are uniformly located within each cell,and 1 000 sample functions of random excitation are simulated for each of these points. The one-step transition time is chosen as

∆t = T = 2π/ω.

4.1 Transient response analysis

The one-step transition probability matrix Q determines the temporal evolution of any initial probability distribution P⁽⁰⁾ . After P⁽⁰⁾ is assigned,the transient response at different moments can be computed by Eq. (5).

The intensity of the Gaussian white noise D is fixed to be 0.006. Let P⁽⁰⁾ be chosen to be uniformly distributed. Some transient marginal PDFs of the system (7) are depicted in Fig. 2. It can be observed in Fig. 2(a) that the marginal PDF has much more probability near the saddle S(1,2) when the evolutionary time t = 1T . As time goes on,the probability of response evolves from the saddle to the two attractors A1 and A2 along the unstable manifold. At the time t = 10T ,the probability distribution of response reaches a stationary state. From the figure,we can also see that all the results obtained from the GCM method agree well with those obtained from the direct Monte Carlo simulation (MCS) method.

Furthermore,let P⁽⁰⁾ concentrate in the elements c₁ or c₂ (the number of cells in which the attractors are located) with the probability 1. The evolutionary process of the transient marginal PDFs of the displacement is shown in Fig. 3. From Fig. 3,we can see that all the results obtained from the GCM method agree well with those from the direct MCS method.

4.2 Stationary response and stochastic P-bifurcation

In this section,the stationary response of system (7) is examined according to the joint PDF obtained from the GCM method. As shown in Fig. 4,the joint PDFs are bimodal. The two peaks correspond to the stable motion of the corresponding deterministic system,i.e.,the attractors A1 and A2. It implies that there are two kinds of more probable motion in the response of the stochastic system and noise induces random transitions between them.

From Fig. 4,we can see that the increase in the noise intensity D has obvious effects on the stationary response of the system. It is found that the two peaks of the joint PDF are better separated at smaller intensity. In such cases,the motion near the attractors of the corresponding deterministic system occupies more probabilities. However,as the noise intensity D increases, the two peaks gradually become broader and even merge in the middle,which indicates that the random transition between the two kinds of more probable motion is more frequent,i.e., the responses of the system become more random.

The change in the number and shape of the peaks of the stationary PDF can be called as the stochastic P-bifurcation. If the effect of the harmonic excitation frequency ω on the stationary PDF is considered,the evolution of the stochastic P-bifurcation can be found. The parameters are taken the same as those in Fig. 4(a) except ω,and the results are given in Fig. 5. When the frequency rate decreases gradually,the peak on the right-side in the bimodal joint PDF becomes flat,and evolves from its position to the left-side. In this process,the evolutionary direction is consistent with the unstable manifold shown in Fig. 1. At last,the stationary PDF becomes unimodal. From the change of the shape of the stationary PDF,we can convince that stochastic P-bifurcation occurs.

5 Conclusions

In the present paper,a system with real-power restoring force under harmonic and Gaussian white noise excitations is investigated. The transient and stationary PDFs are effectively com-puted by the GCM method. Based on the results,the stochastic response analysis is analyzed in combination with the global properties of the noise-free system.

In the transient-state PDFs,it is found that the probability response density gathers in the position of the saddle before evolving to the attractors. For the stationary response,the effect of the noise intensity is examined. Furthermore,as the frequency rate of harmonic excitation decreases,stochastic P-bifurcation occurs. In the evolutionary process,one peak in the stationary PDF becomes flat,and evolves to the other one along the unstable manifold. It implies that the saddle manifold and the unstable manifold of the corresponding deterministic system play important roles in the stochastic response analysis.