Duffing系统随机分岔的全局分析

GLOBAL ANALYSIS OF STOCHASTIC BIFURCATION FOR DUFFING SYSTEMS

应用广义胞映射方法研究了在谐和与随机噪声联合作用下的Duffing系统的随机分岔现象。对于随机Duffing系统,以吸引子形态的突然变化,描述一类随机分岔现象。数值结果表明,随着随机激励强度的逐渐增大,当随机激励强度通过临界值时,随机系统的吸引子与其吸引域边界(吸引域)上的鞍碰撞,发生分岔现象。比较结果表明,在同样的参数区域内,Lyapunov指数均为负值,也就是说,在Lyapunov指数意义下,无法发现这种随机分岔现象。

Since random disturbance or noise always exists in a physical system, the influence of random disturbance on the dynamical behavior, especially bifurcation phenomena of a nonlinear dynamical system has attracted much attention by many researchers. However, the theory of stochastic bifurcation is still in its infancy. In fact, it is much more harder to deal with stochastic bifurcation problems than to deal with deterministic bifurcation problems. The definition of deterministic bifurcation can be based upon the sudden change of topological properties of the portrait of phase trajectories. At present, there are mainly two kinds of definitions for stochastic bifurcation available. One is based on the sudden change of shape of the stationary probability density function——the so-called P-bifurcation (Arnold, 1998); and the other is based on the sudden change of sign of the largest Lyapunov exponent——the so-called D-bifurcation (Arnold, 1998). Lack of certain relationship between the shape variation of the stationary probability density function of the random response and the quantitative variation of the random excitation is the difficulty encountered by P-bifurcation. Lack of efficient and accurate algorithm for calculating Lyapunov exponent is the difficulty encountered by D-bifurcation. Besides, several studies have shown that sometimes these two kinds of definitions may lead to different results (Baxendale, 1986, Meunier et al.,1988, Crauel et al., 1998, Arnold, 1998). For instance, Baxendale (1986) provided an example in which the shape of stationary probability density does not depend on the bifurcation parameter, while the largest Lyapunov exponent changes its sign. On the contrary, Crauel and Flandoli (1998) presented an example in which the stationary probability density function does change its shape from mono-peak one into double-peak one at a critical parameter value, while the largest Lyapunov exponent does not change its sign. Thus, one cannot help thinking about what is really happened for stochastic bifurcation, what is the topological property of a stochastic system, what kind of invariance is suitable for predicting stochastic bifurcation, and so on. Stochastic bifurcation of a Duffing system subject to a combination of a deterministic harmonic excitation and a white noise excitation is studied in detail by the generalized cell mapping method using digraph. It is found that under certain conditions there exist two stable invariant sets in the phase space, associated with the randomly perturbed steady-state motions, which may be called stochastic attractors. Each attractor owns its attractive basin, and the attractive basins are separated by boundaries. Along with attractors there also exists an unstable invariant set, which might be called a stochastic saddle as well, and stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. This definition applies equally well either to randomly perturbed motions, or to purely deterministic motions. This study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.