耦合反应扩散系统的指数稳定性

发布时间：2019-03-28 10:32

【摘要】：在过去的几十年中,耦合系统成为一个热门的研究话题,它在机械、电子以及生物等众多领域中有着广泛的应用。一方面,系统内部各个子系统间的相互作用是存在有限传播速度的,那么时间延迟是一定会存在的。另一方面,耦合系统各个节点的状态还受空间的影响,那么反应扩散是完全不可避免的。所以,为了更准确地对耦合系统的动力学性质进行描述,时间延迟和反应扩散的影响在研究耦合系统动力学行为时都不能被忽视。指数稳定性是耦合系统最重要的动力学性质之一,Razumikhin方法是判别系统稳定性的有效方法,且与Lyapunov泛函方法相比,Razumikhin方法所加的条件限制性更少。有向图可以被用来描述我们所研究的系统,故本文利用图论和Razumikhin方法来研究耦合反应扩散系统的指数稳定性。文章第2章讨论了具有混合延迟的耦合反应扩散系统的指数稳定性。系统中的时间延迟是可变的,而且不只是离散延迟,分布延迟也一并被考虑。应用图论与Razumikhin方法,给出系统的Razumikhin型定理,并由此定理很容易得到Lyapunov型定理。为了更容易验证,通过Lyapunov型定理给出第2章的系数型定理。在这一章最后,一个数值算例验证该章结论的有效性。文章第3章研究了具有延迟和多扩散的耦合反应扩散多组模型的指数稳定性。在研究的模型中,每个组内部以及各个组之间的扩散均受时间延迟和反应扩散的影响。基于图论和Razumikhin方法给出模型的Razumikhin型定理,并通过Razumikhin型定理得到Lyapunov型定理。为了易于验证,给出模型的系数型定理,最后用一个数值算例验证本文结论的有效性。
[Abstract]:In the past few decades, coupling system has become a hot research topic, it has been widely used in many fields, such as machinery, electronics and biology. On the one hand, the interaction between the subsystems in the system has a limited propagation speed, then the time delay is certain to exist. On the other hand, the state of each node in the coupling system is influenced by space, so the reaction diffusion is completely inevitable. Therefore, in order to describe the dynamical properties of coupled systems more accurately, the effects of time delay and reaction diffusion can not be ignored in studying the dynamical behavior of coupled systems. Exponential stability is one of the most important dynamical properties of coupled systems. The Razumikhin method is an effective method to judge the stability of systems. Compared with the Lyapunov functional method, the Razumikhin method has less conditional constraints. The directed graph can be used to describe the system we studied, so we use graph theory and Razumikhin method to study the exponential stability of coupled reaction-diffusion system. In chapter 2, the exponential stability of coupled reaction-diffusion systems with mixed delays is discussed. The time delay in the system is variable, and not only discrete delay, but also distributed delay is considered. Using graph theory and Razumikhin method, the Razumikhin type theorem of the system is given, and it is easy to get the Lyapunov type theorem from this theorem. In order to verify it more easily, the coefficient type theorem in chapter 2 is given by Lyapunov type theorem. At the end of this chapter, a numerical example is given to verify the validity of the conclusion. In chapter 3, the exponential stability of coupled reaction-diffusion model with delay and multi-diffusion is studied. In the model, diffusion within and between each group is affected by time delay and reaction diffusion. Based on graph theory and Razumikhin method, the Razumikhin type theorem of the model is given, and the Lyapunov type theorem is obtained by the Razumikhin type theorem. In order to be easy to verify, the coefficient type theorem of the model is given. Finally, a numerical example is used to verify the validity of the conclusion in this paper.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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