Mathematical Sciences: Conley Index Theory and Applications

点击翻译按钮获取中文摘要

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. ***