Nonlinear Dynamics in Heterogeneous and Random Systems

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NSF Award Abstract - DMS-0072311Mathematical Sciences: Nonlinear Dynamics in Heterogeneous and Random SystemsAbstract0072311 KuskeThe project covers four areas of research, which all use a combination of asymptotics and computation. The first goal is to understand how small noise alters transitions in nonlinear dynamics, focusing on interface dynamics, transitions in bursters, and oscillations in lasers with delayed feedback. This research also contributes to a broad characterization of the effects of noise in nonlinear systems, useful for effective modeling. The second objective is to predict and understand localized behavior in coupled oscillators, coupled laser arrays, calcium oscillations, and structural dynamics. The strength of the localization can depend on competing effects in a non-uniform way. The third objective is to understand the effects of heterogeneity on the dynamics of patterns in reaction-diffusion problems, using both envelope equation approximations and computations of the full model. The fourth goal is developing new methods for pricing financial derivatives. To price a variety of options, we combine methods for moving boundaries in partial differential equations with singular perturbation methods.The overall goal of the project is to develop asymptotic theory and methods for mathematical models of heterogeneous complex systems. Such methods are often combined with computations in order to completely analyze these models. When including disorder or inhomogeneities in a mathematical model, we try to answer the following questions: Does the heterogeneity cause qualitative changes in the behavior of the system? Is the model highly sensitive to physical variations or computational inaccuracies? Is the model realistic, and can similar models be used as accurate descriptions or predictors for the "real world" system? What are the deficiencies of the model, and how might it be improved? In this project we focus on mathematical models in which heterogeneity plays a significant role, including models of laser dynamics, chemical reactions, calcium concentrations in cells, neuronal dynamics, engineering structures, and financial products.