Mathematical Sciences: Conley Index Theory and Applications

9302970 Mischaikow正在使用Conley指数来研究进化方程的全局动力学。Mischaikow试图回答的第一个问题是给定一组参数化的微分方程在所有参数值下系统共有的最小动态结构是什么。对于诸如Cahn-Hilliard， Ginzburg-Landau, Fitz Hugh-Nagumo，带负反馈的微分延迟等方程，目标是导出非平凡模型流和从感兴趣系统的全局吸引子到模型流的连续抛射。对于其他动力系统，如Henon映射、Lorenz方程、非单调循环反馈系统，目标是从不变集到有限符号集上的子移动力学的半共轭性。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计算受扰系统的康利指数（从而获得有关动力学的信息）。考虑的问题包括薄域和快慢系统。大多数数学模型不是从第一原理推导出来的，因此，是基于对物理系统的启发式、直觉理解和近似。这意味着不可能精确地知道模型中使用的适当参数值甚至非线性。因此，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具有鲁棒性是很重要的。Mischaikow的工作目标是开发适用的数学技术来识别这种健壮的动力学结构。* * *

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. ***