Stability of dynamical systems
Stability of dynamical systems
复制标题
DOI：
10.1088/0951-7715/1/1/005
复制
发表时间：
1988-02
期刊：
影响因子：
1.7
通讯作者：
A. N. Michel;Derong Liu;Ling Hou
中科院分区：
文献类型：
--
作者：
A. N. Michel;Derong Liu;Ling Hou
提出了常微分方程稳定性的一个新定义，作为结构稳定性的替代。它特别针对耗散非线性系统，包括那些与混沌或奇怪的吸引子。定义如下。给定一个定向流形X上的向量场nu，且给定nu <0，设u为nu具有nu-扩散的福克-普朗克方程的定态。在X是紧的情况下证明了u的存在性、唯一性和全局吸引性（在非紧的情况下，u的存在需要一个合适的边界条件）.向量场被定义为等价的或稳定的，这取决于它们的稳态是否是。一个类似的理论是发展的复同态。新定义与结构稳定性相比有许多优点。稳定系统是稠密的，因此大多数奇异吸引子是稳定的，包括非双曲吸引子。这种等价性将梯度系统的Thom分类推广到非梯度系统。该理论与应用密切相关，因为稳态u是nu流吸引子上的测度的一种平滑，因此在数值和物理实验中，u可以用来模拟具有平滑误差的数据。
A new definition of the stability of ordinary differential equations is proposed as an alternative to structural stability. It is particularly aimed at dissipative nonlinear systems, including those with chaos or strange attractors. The definition is as follows. Given a vector field nu on an oriented manifold X, and given epsilon <0, let u be the steady state of the Fokker-Planck equation for nu with epsilon -diffusion. The existence, uniqueness and global attraction of u is proved in the case when X is compact (in the non-compact case a suitable boundary condition on nu is required for the existence of u). Vector fields are defined to be equivalent, or stable, according to whether their steady states are. A similar theory is developed for diffeomorphisms. The new definition has a number of advantages over structural stability. Stable systems are dense, and therefore most strange attractors are stable, including non-hyperbolic ones. The equivalence extends the Thom classification of gradient systems to non-gradient systems. The theory is closely related to applications, because the steady state u is an epsilon -smoothing of the measure on the attractors of the flow of nu , and therefore in numerical and physical experiments u can be used to model the data with epsilon -error.