Mathematical Sciences: Conley Index Theory and Applications

9302970米夏科夫正在使用康利指数来研究演化方程的全球动力学。Mischaikow试图回答的第一个问题是：给定一个参数化微分方程组，在所有参数值下，系统的最小动态结构是什么。对于Cahn-Hilliard，Ginzburg-Landau，Fitz Hugh-Nagumo，具有负反馈的微分时滞等方程，目标是从感兴趣系统的全局吸引子导出非平凡的模型流和连续的满足点到模型流上。对于其他动力系统，如Henon映射、Lorenz方程、非单调循环反馈系统，目标是在有限符号集上获得从不变集到子移位动力学的半共轭。第二个问题涉及奇异摄动问题。在这种情况下，目标是仅使用奇异系统的动力学来计算扰动系统的Conley指数(从而获得关于动力学的信息)。正在考虑的问题包括薄域和速度较慢的系统。大多数数学模型不是从最初的原理得出的，因此是基于启发式的、直观的理解和物理系统的近似。这意味着不可能准确地知道模型中使用的适当参数值，甚至是非线性。因此，重要的是，在数学系统中观察到的定性动态结构对于非线性和参数的变化是鲁棒的。米沙科夫工作的目标是开发出能够识别这种稳健的动力学结构的适用的数学技术。***

9302970 Mischaikow Mischaikow is using the Conley index to study the global dynamics of evolution equations. The first question which Mischaikow is trying to answer is: given a parameterized family of differential equations what minimal dynamic structure is common to the system at all parameter values. For equations such as Cahn-Hilliard, Ginzburg-Landau, Fitz Hugh-Nagumo, differential delay with negative feedback and others the goal is to derive nontrivial model flows and continuous surjections from the global attractors of the systems of interest onto the model flows. For other dynamical systems such as the Henon map, the Lorenz equations, non-monotone cyclic feedback systems the goal is to obtain a semi-conjugacy from an invariant set to subshift dynamics on a finite set of symbols. The second question involves singular perturbation problems. The goal in this case is to compute Conley indices for the perturbed system (and thereby gain information about the dynamics) using only the dynamics of the singular system. The problems being considered include thin domains and fast slow systems. Most mathematical models are not derived from first principals, and hence, are based on heuristics, intuitive understandings, and approximations of the physical system. This means that it is impossible to know precisely the appropriate parameter values or even nonlinearities used in the model. Therefore, it is important that the qualitative dynamic structures observed in the mathematical system be robust with respect to changes in the nonlinearities and parameters. The goal of Mischaikow's work is to develop applicable mathematical techniques which can identify such robust dynamics structures. ***