Mathematical Sciences: Homoclinic and Heteroclinic Bifurcations: Theory and Applications

9404070邓，应用数学家的传统角色是获取一个真正的系统，并找出它的性质。然而，只有知道可能性是什么，才能发现特定系统的性质，而这些可能性往往只有通过一般的抽象理论才能揭示出来。实践与理论最好携手共进。在这份提案中，提出了三个项目。它们都涉及在真实系统中产生复杂和混乱结构的基本机制。理论结果有助于更好地理解化学反应系统、胰岛β细胞的胰岛素分泌以及沿神经轴突的电脉冲的实验和数值结果。它们对这些系统未来的实验和实际应用也具有重要的指导意义。提出了三个项目。第一个方案涉及一种退化的异宿回路分叉，并将其应用于通过填充塔的两相流中脉冲波的存在。第二个项目涉及胰腺β细胞电活动的Rinzel-Terman模型中的轨道翻转同宿分叉。这种分叉与膜电位从连续峰向间歇峰的转变有关。第三个项目是关于寻找双稳态反应扩散方程组中扩散参数的分支值的策略和数值技术。这些值是这样的，对应于行进的前波和后波的异宿环路改变了其扭曲类型，这通常是混沌动力学产生的信号。

9404070 Deng The traditional role of the applied mathematician is to take a real system and find out what its properties are. However, the properties of the specific system cannot be discovered unless one knows what the possibilities are, and the possibilities are often revealed only by the general abstract theory. Practice and theory progress best hand-in-hand. Three projects are proposed in this proposal. They all deal with basic mechanisms by which complex and chaotic structures arise in real systems. The theoretical results are useful in giving better understanding to experimental and numerical results from a system of chemical reaction, insulin secretion in pancreatic beta cells, and electrical pulses along nerve axon. They are also important in guiding future experiments and practical applications of these systems. Three projects are proposed. The first project concerns a type of degenerate heteroclinic loop bifurcation with application to the existence of pulsing waves in a two-phase flow through a packed column. The second project concerns the orbit-flip homoclinic bifurcation in the Rinzel-Terman model for the electrical activity of pancreatic beta cells. The bifurcation is related to the transition from continuous spiking to intermittent spiking of the membrane potential. The third project concerns strategy and numerical technique for locating bifurcation values of a diffusion parameter in a system of bistable reaction diffusion equations. These values are such so that a heteroclinic loop corresponding to traveling front and back waves changes its twist type, which often signals the generation of chaotic dynamics.