Applied dynamical systems and asymptotic analysis

本文的研究是两个领域的结合：应用动力系统和渐近分析，以及它们之间的相互作用。未来五年的主要研究课题包括行波和速度选择的研究，表面图案的形成（尖峰，冲击和斑点 * 图案）和渐近分析（例如，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。 最近的进展 * 对空间扩散和时间 * 演化的联合作用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。 对这些流行系统的非线性动力学（例如行波波前、有界域中稳态的稳定性）进行定性和定量分析是我未来研究的主要目的。我们特别感兴趣的是行波模式的存在性和波速选择机制。2)用表面反应扩散方程模拟斑点斑图的形成和亚稳态运动分析。各种物理、生物和化学 * 模型中的图案形成和指数缓慢内层运动 *（亚稳态运动）的 * 表征将是我未来研究的另一个主题。我的目标是对表面上的反应扩散系统中斑点层溶液的形成和运动进行定性和定量分析。3)曲面上奇摄动 * 偏微分方程解的分析与逼近。 我将研究曲面上奇摄动偏微分方程边值问题解的逼近。将提供近似解的误差界，这将为过去参考文献中的一些形式分析提供严格的论据。 从理论上讲，这一建议表明了应用动力系统和渐近分析理论的新发展。 这项研究在生物学和物理学等其他科学的应用中也很重要。通过该提案，旨在建立一流的研究团队在我们的大学，这也大大有助于加拿大数学社会。

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.**