格微分方程和状态依赖时滞微分方程的动力学研究

发布时间：2018-06-06 20:01

本文选题：格微分方程 + Lyapunov-Schmidt约化　；参考：《湖南大学》2015年博士论文

【摘要】：本文主要研究格微分方程和状态依赖时滞微分方程的动力学行为.通常,格微分系统是由定义在离散空间上的无穷多个常微分方程耦合而成的系统.随着计算机的飞跃发展,数值计算已经成为格微分方程的重要应用背景之一,同时在图像处理,化学反应,材料科学和生物学等领域都应用到格微分模型.格微分方程一方面能更确切地描述实际问题,另一方面,它比连续的偏微分方程具有更丰富的动力学行为.状态依赖时滞微分方程的理论体系目前还不够完善,关于这一方面的理论及其应用著作也相对比较少.随着各个学科的蓬勃发展,越来越多的状态依赖时滞微分方程被用来描述物理,自动控制,神经网络,传染性疾病,人口增长和细胞繁殖等领域的动力学过程.因此,对状态依赖时滞微分方程的研究具有重要的理论和实际意义.本文主要研究了一类二维格微分方程的波列问题,以及两类状态依赖时滞微分模型的动力学行为.本文的主要内容如下:第一,我们研究一类二维格微分方程的波列解存在性和分支模式.由于系统的波动方程可看作一个既具有超前项又具有时滞的微分方程,不能运用动力系统的半流理论来进行研究.我们通过Lyapunov-Schmidt约化,将该方程约化为一个有限维空间上的具有某种对称性的分岔方程,且保持原系统的哈密尔顿性.应用不变理论和奇异性理论,我们在非共振和共振p:q两种情形下得到了平衡点附近的小振幅解.我们分析了传播方向θ的影响,并发现当tanθ为有理数时,原二维格系统可转化为一维格系统,且原系统的结论可直接应用于对应的一维格系统,从而得到某些一维格微分系统的波列存在性和分支模式.第二,我们研究状态依赖时滞Nicholson飞蝇模型的动力学行为.首先,在对时滞函数和方程中的参数作适当的假设下,分析了解的一些基本性质,包括解的存在唯一性,关于初值的连续依赖性,振荡性等.接着,我们构造了一个适当的紧状态空间,使得系统的解构成一个连续半流.通过引进离散的Lyapunov泛函来分析了慢振荡解,我们发现所有全局慢振荡解所形成的集合构成半流的一个全局吸引子.然后,通过线性化过程和谱分析,得到了系统平衡态的局部动力学性质,给出了正平衡态附近的局部不稳定流形.最后,在局部不稳定流形中,选取正平衡态的一个充分小的邻域,将它延拓为一个全局不稳定流形.通过分析该全局不稳定流形的零点集,我们证明了慢振荡周期解的存在性,且它正好构成全局不稳定流形的边界.第三,我们利用不动点定理研究了一类二阶状态依赖时滞微分方程的慢振荡周期解存在性.首先,我们得到解的一些基本性质,包括初值问题解的存在唯一性和关于初值的连续依赖性.然后,详细地分析了系统的慢振荡解.基于前面的结论,我们构造了一个紧集和一个后继映射,运用不动点定理和平衡态的喷射性证明慢振荡周期解的存在性.
[Abstract]:In this paper, we mainly study the dynamic behavior of differential equations and state dependent delay differential equations. Generally, the lattice differential system is a system which is coupled to infinitely many ordinary differential equations defined on the discrete space. With the rapid development of the computer, the numerical calculation has become one of the important application backgrounds of the lattice micro equation. The lattice differential model is applied in such fields as treatment, chemical reaction, material science and biology. The lattice differential equation can describe the actual problems more accurately on the one hand, and on the other hand, it has more abundant dynamic behavior than the continuous partial differential equation. The theory system of state dependent delay differential equation is not perfect at present. With the vigorous development of various disciplines, more and more state dependent delay differential equations are used to describe the dynamic processes in the fields of physics, automatic control, neural networks, infectious diseases, population growth and cell reproduction. Therefore, the study of state dependent delay differential equations The main contents of this paper are as follows: first, we study the existence and bifurcation of the wave solutions of a class of two-dimensional lattice differential equations. We study the wave solution existence and bifurcation of a class of two-dimensional lattice differential equations. The process can be considered as a differential equation with both the preterm and time delay. We can not use the semi flow theory of the dynamic system to study it. Through the reduction of Lyapunov-Schmidt, we reduce the equation to a bifurcation equation with some symmetry in a finite dimensional space, and hold the Hamilton property of the original system. And singularity theory, we obtain the small amplitude solution near the equilibrium point in two cases of non resonance and resonance p:q. We analyze the influence of the propagation direction theta, and find that when the tan theta is rational, the original two wig system can be transformed into a one wig system, and the original system can be directly applied to the corresponding one wig system, and thus the result can be obtained. Second, we study the dynamic behavior of the state dependent time-delay Nicholson fly model. First, we analyze some basic properties, including the existence and uniqueness of the solution and the continuous dependence of the initial values, under the proper assumption of the parameters in the delay function and the equation. Oscillation and so on. Then, we construct a proper tight state space so that the solution of the system constitutes a continuous half flow. By introducing the discrete Lyapunov functional, we analyze the slow oscillation solution. We find that all the sets formed by the global slow oscillation solutions constitute a full local attractor for the semi flow. Then, the linearization process and the spectral fraction are used. In this paper, the local dynamic properties of the equilibrium state of the system are obtained, and the local unstable manifolds near the positive equilibrium state are given. Finally, in the local unstable manifold, a sufficient small neighborhood of the positive equilibrium state is chosen to extend it into a global unstable manifold. By analyzing the zero point set of the global unstable manifold, we prove the slow vibration. The existence of the periodic solution, and it just constitutes the boundary of the global unstable manifold. Third, we use the fixed point theorem to study the existence of a class of slow oscillation periodic solutions for a class of two order state dependent delay differential equations. First, we get some basic properties of the solution, including the existence and uniqueness of the solution of initial value problems and the continuity of initial values. Then, the slow oscillation solution of the system is analyzed in detail. Based on the previous conclusion, we construct a compact set and a successor mapping, and use the fixed point theorem and the jetting of the equilibrium state to prove the existence of the periodic solution of the slow oscillation.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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