Applied dynamical systems and asymptotic analysis

本课题的研究是应用动力系统和渐近分析两个领域的结合，以及它们之间的相互作用。未来五年的主要研究方向包括行波和速度选择的研究，表面上的模式形成（尖峰、激波和斑*模式）和渐近分析（如奇异摄动和边界层理论）。研究范围和目标是：1)分析反应扩散模型的波传播模式和波速选择机制。最近关于空间扩散和时间演化的联合效应的研究进展导致了一些有趣的反应扩散方程。这些模型在物理、生物和化学科学中得到了广泛的研究。这些流行系统的非线性动力学的定性和定量分析（例如行波前，它们在有界域中的稳态稳定性）是我未来研究的主要目的。特别地，我们对行波模式的存在和波速选择力学感兴趣。*2)基于表面反应扩散方程的斑点模式形成和亚稳运动分析。在各种物理、生物和化学模型中，模式形成和指数级慢内层运动（亚稳态运动）的表征将是我未来研究的另一个主题。我的目标是对表面反应扩散系统中出现的斑点层溶液的形成和运动进行定性和定量分析。*3)曲面上奇异摄动偏微分方程解的分析与近似。我将研究曲面上奇摄动*偏微分方程边值问题的近似解。给出了近似解的误差范围，为以往文献中的一些形式分析提供了严格的论证。***从理论上讲，这一建议建立了应用动力系统理论和渐近分析的新发展。该研究在生物学和物理学等其他科学领域的应用也很重要。通过提案，意在我校建立一支一流的研究团队，这对加拿大数学界也有很大的贡献

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.**