【作者】 张若军；

【导师】 王林山；

【作者基本信息】 中国海洋大学 ， 海洋信息探测与处理， 2009， 博士

【摘要】 随着科学技术的快速发展,在物理、力学、控制理论、生物学、医学和经济学等自然科学及边缘学科的研究领域提出了大量由时滞动力方程(也称时滞动力系统)所描述的具体数学模型,因而对时滞动力系统进行研究在理论和实际应用方面都有重要意义。时滞动力系统属于非线性动力学的分支。经典非线性动力学是以扰动、渐近分析的方法研究弱非线性弱耦合的系统。而现代非线性动力学与经典非线性动力学不同,研究的是系统的定性与定量的变化规律,其使用的方法是精确方法,所研究的系统具有强非线性性,其研究对象主要包括稳定性、周期解、吸引子以及分叉、混沌、孤立子等新的现象,其主要任务是探索非线性科学的复杂性。本文对著名的Duffing型, Liénard型方程以及区间细胞神经网络、反应扩散神经网络、联想记忆神经元模型进行了研究和推广,考虑时滞对系统稳定性的影响,运用多种方法研究了几种时滞泛函微分方程所表示的模型的全局动力行为,包括:周期解的存在性,鲁棒稳定性,分叉等,并利用Matlab求解泛函微分方程和对神经网络进行了仿真。本文共由6章组成,主要内容如下:首先是第1章绪论,概述了时滞动力方程的研究背景和发展状况,并简要介绍了本文的主要工作。第2章,利用重合度理论(Mawhin延拓定理)、不等式理论以及各种变换技巧研究了推广的时滞Duffing型、Liénard型方程,考虑其周期解的存在性问题,得到了某些充分条件,所得结果推广和改进了相关文献的结果。第3章,一方面,在有限区间情况下,采用拓扑度理论和建立适当的Liapunov-函数方法,讨论区间细胞神经网络的渐近鲁棒稳定性问题。另一方面,在无穷区间情况下,在对无穷区间上的Lebesgue-Stieltjes积分的连续性、可微性进行研究的基础上,应用新的分析技巧和构建适当的带有Lebesgue-Stieltjes积分的Liapunov泛函,讨论区间细胞神经网络的全局指数鲁棒稳定性。两种情况均得到了简单有效的判别准则并给出了实例。所得结论有助于系统结构稳定性分析。第4章,通过截断函数和截断方程,给出了S-分布时滞的反应扩散神经细胞网络模型(RDCNNs)具有全局指数稳定性的平衡点唯一存在的充分性条件,去掉有关文献中要求信号函数f j和g j, j = 1, 2,,n,有界性、单调性和可微性的苛刻条件。给出实用有效的M ?矩阵判断S-分布时滞RDCNNs稳定性的代数方法。所的结果通过实例得到了验证。第5章,研究了具有非单调动力系统的神经元联想记忆模型,其中的输出函数不是sigmoid函数而是非单调的。通过分析非单调系统的分叉相图,获得基于特征方程和Hassard技巧的渐近稳定原则。利用正规型理论和中心流形定理分析了Hopf分叉的稳定性和方向。最后,利用数值模拟对时滞变化引起的相图的变化进行了仿真。所得到的网络具有一定的应用价值。最后,对本文所研究的内容和主要结果进行了总结,并对研究工作的前景做了展望。更多还原

【Abstract】 With the increasing development of science and technology, many mathematical models which are described by delay dynamical equations(delay dynamical systems)are proposed in the field of natural science and edging field including physics, mechanics, control theory, biology, medicine and economics. Therefore, the study of delay dynamical systems is important in theory and practical application.Delay dynamical systems belong to nonlinear dynamics. Classic nonlinear dynamics focus on weakly nonlinear and weakly coupled systems by the methods of disturbance and asymptotic analysis. While modern nonlinear dynamics is different from classic one, it studies the rules of the qualitative and quantitative change of systems. The research methods are precision. The systems are strongly nonlinear. The research objectives are stability, periodic solution, attractor, bifurcation, chaos, and solitons. The aim is to discovery the complexity of nonlinear dynamics.In this paper, the author studies further on some mathematics models.Considering the effecting of time delay to the stability of the systems and applying many methods, the author investigates global dynamical behaviors of several kinds delay functional differential equations. The author studys the existence of periodic solutions, robust stability, bifurcation, and introduce the solution of functional differential equations and neural network simulation with Matlab.This paper is composed of six chapters and main results are described as follows: First, the author introduces the historical background and the recent development of delay dynamic systems and neural networks. Moreover, the main results of this paper are also briefly introduced.Second, by applying coincidence degree theory,the existence of periodic solution of Duffing-type and generalized Liénard-type equation with delay is investigated, and some sufficient criteria are established,which improve and generalize some known results.Third, one hand, applying topological degree theory and constructing suitable Liapunov functional, the author investigates asymptotic robust stability of interval cellular neural networks with S-type distributed delays on finite intervals. The other hand, on the base of studying of continuity and differentiability of Lebesgue-Stieltjes integration on infinitely interval, the author investigates global exponential robust stability of interval cellular networks with S-type distributed time delay by applying the new analysis techniques and constructing suitable Liapunov functional with Lebesgue -Stieltjes integration. In two cases, the convenient criteria and examples are presented. The conclusions are useful to analyze the stability of systems.Fourth, via the intercept functions and intercept equation, the author presents some sufficient conditions for the existence of a unique globally exponentially stable equilibrium of reaction-diffusion cellular neural networks (RDCNNs) with S-type distributed time delay, in which the boundedness, monotonicity and differentiability of signal functions f j and g j,j = 1,2,,n defined on R are not required. The results are verifiable and the method of M-matrix is more practical.Fifth, the associative memory of neurons model with nonmonotone dynamics is considered, where the output function is not sigmoid but nonmonotonic. After analyze the bifurcating phase plot of the nonmonotonic system, the author obtains the asymptotically stability criteria based on the characteristic equation and technique in Hassard. The stability and direction of the Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Finally, numerical simulations have been used to demonstrate the change of phase plot with respect to the time delay. The networks studied above have value in application to some extend.Finally, the studying contents and the main results of this paper are briefly summarized, furthermore, some research prospects are also proposed.更多还原