Applied dynamical systems and asymptotic analysis

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The research in this proposal is a combination of two fields:*applied dynamical systems and asymptotic analysis, as well as the*interaction between them. Main research topics in future five years include the study of traveling waves and speed selection, pattern formation(spike, shock and spot*patterns) on surfaces and asymptotic analysis (e.g., singular perturbation and boundary layer theory). The research scope and objectives are: 1) Analysis of wave-propagation patterns and wave-speed selection mechanics for*reaction diffusion models. The recent progress*towards the combined effect of spatial diffusion and temporal*evolution leads to some interesting reaction diffusion equations. These models have been extensively studied in physical, biological and chemical*sciences. The qualitative and*quantitative analysis of nonlinear dynamics (e.g. traveling*wavefronts, stability of their steady state in a bounded domain)*of these popular systems are the main purpose*of my research in the future. In particular, we are interested in the existence of traveling wave-patterns and the wave-speed selection mechanics.*2) Analysis of pattern formations and metastable-motion of spot patterns*modeled by reaction diffusion equations on surfaces. The*characterization of pattern formations and exponentially slow internal layer motion*(metastable-motion) in various physical, biological and chemical*models will be another subject of my future research. My goal is*to give qualitative and quantitative analysis of formation and movement of spot layer solutions that exhibit in*reaction-diffusion systems on surfaces.*3) Analysis and approximation of solutions to singularly perturbed*partial differential equations on surfaces. I will study the approximation of solutions to boundary value problems for singularly perturbed*partial differential equations on surfaces. Error bounds of the approximate solutions will be provided and this will give rigorous arguments to some formal analysis in past references.*** Theoretically, this proposal demonstrates the establishment of new developments in the theory of applied dynamical systems and asymptotic analysis. The research is also important in the applications to other sciences such as biology and physics. Through the proposal, it is intended to establish a first-class research team in our university and this also contributes greatly to the Canadian mathematical society.**