Dynamical Systems and Singular Perturbation Theory for Multiscale Reaction-Diffusion Systems

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This research project encompasses a series of critical mathematical and scientific questions for multiscale problems arising in the fields of pattern formation, chemistry and combustion, neuroscience, electrical engineering, and multi-particle systems. The first project is on pattern formation and analyzes the dynamics and stability of fronts, pulses, and spots in paradigm reaction-diffusion systems. The second project studies model reduction methods used in complex multiscale chemical reactions, biochemical networks, and combustion by incorporating also the effects of diffusion. The third project involves a completely new class of solutions, known as torus canards, found in models from neuroscience. These solutions help understand the transitions between periodic spiking and bursting. The fourth project will focus on ways to model and analyze the impacts of cut-offs on the dynamics of fronts. The project involves graduate students and postdoctoral fellows in the research, as well as collaborations with scientists at national laboratories. This research project addresses a series of questions concerning multiscale problems in pattern formation, chemistry and combustion, neuroscience, electrical engineering, and multi-particle systems. In the pattern formation project, paradigm reaction-diffusion systems will be studied. The goals are to develop new analytical techniques and mathematical theory for determining the boundaries of the stable pattern-forming regimes, analyzing the stability of semi-strong pulse interactions, modeling the scattering of pulses in 1-D systems and spots in 2-D systems, extending renormalization group methods for stability of modulating pulses, and predicting the dynamic bifurcations of pulses and fronts. The second project centers on accurate model reduction methods for large-scale combustion, chemical, and biochemical systems exhibiting multiple time scales. The goals are to analyze, develop, and improve cutting-edge model reduction methods for finding the low-dimensional manifolds that govern the effective system dynamics in the presence of diffusion. In the third project, the new phenomena of torus canards and canards in partial differential equations will be investigated. A theory of generic torus canards will be developed for fast-slow systems with multi-dimensional fast and slow variables. Known to exist in many neuroscience models, such as the Hindmarsh-Rose equations, the Morris-Lecar-Terman model, the Wilson-Cowan-Izhikevich system, and the forced van der Pol equation, torus canards are critical in the transition regimes between tonic spiking and bursting. A detailed study will also be carried out of the new bursting rhythms known as amplitude-modulated bursting rhythms. The fourth project will study the impacts of cut-offs on the reaction terms, introduced to accurately model regions of low particle densities, on the speeds, shapes, and stability of propagating fronts. A series of important problems related to fourth-order models, two-dimensional space dynamics, front initiation, and front pre-cursors will be studied.