脉冲微分系统周期解存在性与稳定性研究
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摘要
第一章,简述脉冲微分方程在种群系统与生命科学及神经网络系统中的研究发展状况；介绍本文问题产生的背景和本文的主要工作以及一些预备知识。
在第二章的第一节,利用Mawhin连续定理研究一类有扩散、时滞、周期环境以及脉冲影响的n-斑块竞争系统正周期解的存在性,然后利用Lyapunov泛函方法和一些分析技巧讨论得到了该系统全局吸引的充分条件,最后讨论了所得结果的一些应用；第二节利用脉冲微分方程比较原理、一个重要引理和一些分析技巧,讨论了一类具有脉冲扩散与功能反应的捕食-食饵系统边值周期解的存在性、全局吸引性以及系统持续生存的充分条件。
第三章利用Floquet理论和脉冲微分方程的微小扰动技巧,考虑了一类具有Ivlev-型功能反应且食饵有病的捕食-食饵系统在有杀虫控制、投放天敌等脉冲影响下的综合控制(IPM控制)策略问题,得到食饵灭绝周期解存在且全局吸引以及系统持续生存的充分条件；通过数值分析进一步说明了生态系统动力学行为的复杂性。
第四章利用Mawhin连续定理讨论得到了一类具有变时滞与脉冲的微分方程周期解的存在性,利用一个重要引理与一些分析技巧,得到周期解的唯一性、全局吸引性以及该系统持续生存的充分条件；所得结果推广或改进了一些已有成果。
第五章首先构造了一类具有变时滞与脉冲影响的细胞神经网络系统,利用Young不等式并通过构造合适的Lyapunov函数,讨论该模型存在周期解及指数稳定的充分条件；然后讨论一类有变时滞与脉冲影响的BAM神经网络系统,利用Mawhin连续定理与矩阵分析技巧得到该系统存在周期解的充分条件,利用Lyapunov泛函方法讨论了周期解的稳定性,得到了系统存在周期吸引子的充分条件；最后介绍了所得结果的一些应用。
This Ph.D. thesis is divided into five chapters and main contents are as follows:
In Chapter 1, a survey to the developments of the existence and stability of periodic solutions of impulsive differential equations, impulsive predator-prey system and impulsive neural networks is given. Then the background of problems, the main results of this dissertation are introduced and some preliminaries are also summarized.
In Chapter 2, firstly a class of one predator n-patch-prey model with diffu-sion, impulsive and delays are established. By using Mawhin continuation theorem, Lyapunov functional method and some analysis techniques, the sufficient condi-tions ensuring the existence of positive periodic solutions and global attractivity are obtained in section 2.1. Secondly, by using impulsive comparison theorem, an important lemma and some analysis techniques, the sufficient conditions of the existence of boundary periodic solution, global attractivity and the permanence for a predator-prey system with impulsive diffusion and functional response are obtained in section 2.2.
In Chapter 3, an impulsive predator-prey system with Ivlev-type functional response and disease in the prey is considered. By using Floquet theory and Lyapunov functional, the existence and global attractivity of the prey-extinction periodic solution and the permanence of the system are studied. Finally, the complexity of the dynamical behaviors for the predator-prey system is showed by numerical analysis. It is important and valuable for IPM control.
In Chapter 4, by using Mawhin continuation theorem, an important lemma and some analysis techniques, the sufficient conditions ensuring the existence of the periodic solution, global attractivity and the permanence of a generalized delayed differential system with impulses are obtained. The main results generalize or improve the previously known results.
In Chapter 5, a class of neural networks with delays and impulses is considered in section 5.1. By using Young inequality and constructing Lyapunov functional, Sufficient conditions ensuring the existence and exponential stability of periodic solutions are obtained. Next, a class of BAM neural networks with time-varying delays and impulses is considered in section 5.2. The existence and stability of periodic solution are studied by using continuation theorem, M-matrix theory and Lyapunov functional. Finally, some applications and examples are given to show the effectiveness of the main results.
在第二章的第一节,利用Mawhin连续定理研究一类有扩散、时滞、周期环境以及脉冲影响的n-斑块竞争系统正周期解的存在性,然后利用Lyapunov泛函方法和一些分析技巧讨论得到了该系统全局吸引的充分条件,最后讨论了所得结果的一些应用；第二节利用脉冲微分方程比较原理、一个重要引理和一些分析技巧,讨论了一类具有脉冲扩散与功能反应的捕食-食饵系统边值周期解的存在性、全局吸引性以及系统持续生存的充分条件。
第三章利用Floquet理论和脉冲微分方程的微小扰动技巧,考虑了一类具有Ivlev-型功能反应且食饵有病的捕食-食饵系统在有杀虫控制、投放天敌等脉冲影响下的综合控制(IPM控制)策略问题,得到食饵灭绝周期解存在且全局吸引以及系统持续生存的充分条件；通过数值分析进一步说明了生态系统动力学行为的复杂性。
第四章利用Mawhin连续定理讨论得到了一类具有变时滞与脉冲的微分方程周期解的存在性,利用一个重要引理与一些分析技巧,得到周期解的唯一性、全局吸引性以及该系统持续生存的充分条件；所得结果推广或改进了一些已有成果。
第五章首先构造了一类具有变时滞与脉冲影响的细胞神经网络系统,利用Young不等式并通过构造合适的Lyapunov函数,讨论该模型存在周期解及指数稳定的充分条件；然后讨论一类有变时滞与脉冲影响的BAM神经网络系统,利用Mawhin连续定理与矩阵分析技巧得到该系统存在周期解的充分条件,利用Lyapunov泛函方法讨论了周期解的稳定性,得到了系统存在周期吸引子的充分条件；最后介绍了所得结果的一些应用。
This Ph.D. thesis is divided into five chapters and main contents are as follows:
In Chapter 1, a survey to the developments of the existence and stability of periodic solutions of impulsive differential equations, impulsive predator-prey system and impulsive neural networks is given. Then the background of problems, the main results of this dissertation are introduced and some preliminaries are also summarized.
In Chapter 2, firstly a class of one predator n-patch-prey model with diffu-sion, impulsive and delays are established. By using Mawhin continuation theorem, Lyapunov functional method and some analysis techniques, the sufficient condi-tions ensuring the existence of positive periodic solutions and global attractivity are obtained in section 2.1. Secondly, by using impulsive comparison theorem, an important lemma and some analysis techniques, the sufficient conditions of the existence of boundary periodic solution, global attractivity and the permanence for a predator-prey system with impulsive diffusion and functional response are obtained in section 2.2.
In Chapter 3, an impulsive predator-prey system with Ivlev-type functional response and disease in the prey is considered. By using Floquet theory and Lyapunov functional, the existence and global attractivity of the prey-extinction periodic solution and the permanence of the system are studied. Finally, the complexity of the dynamical behaviors for the predator-prey system is showed by numerical analysis. It is important and valuable for IPM control.
In Chapter 4, by using Mawhin continuation theorem, an important lemma and some analysis techniques, the sufficient conditions ensuring the existence of the periodic solution, global attractivity and the permanence of a generalized delayed differential system with impulses are obtained. The main results generalize or improve the previously known results.
In Chapter 5, a class of neural networks with delays and impulses is considered in section 5.1. By using Young inequality and constructing Lyapunov functional, Sufficient conditions ensuring the existence and exponential stability of periodic solutions are obtained. Next, a class of BAM neural networks with time-varying delays and impulses is considered in section 5.2. The existence and stability of periodic solution are studied by using continuation theorem, M-matrix theory and Lyapunov functional. Finally, some applications and examples are given to show the effectiveness of the main results.
引文
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