【作者】 陈丽；

【导师】 张伟年；

【作者基本信息】 四川大学 ， 应用数学， 2006， 博士

【摘要】 动力系统就是要研究一个确定性系统的状态变量随时间变化的规律，根据系统变化的规律可分为由微分方程描述的连续动力系统和由映射迭代揭示的离散动力系统，大量的物理、力学、生物学以及天文学问题的数学模型都是有连续的和离散的迭代过程描述的．迭代是自然界乃至整个人类生活中的一种普遍现象．迭代方程是包含未知映射迭代的等量形式，它在自然界中有许多重要的应用．例如，在描述倍周期分岔普适性时的费根鲍姆(Feigenbaum)现象、微分方程中的不变流形、Hamilton系统中的不变环面和不变曲线、正规形问题等都可归结为对迭代方程的研究，迭代方程已成为与微分方程、差分方程、积分方程及动力系统紧密相关的重要的数学方程形式，受到众多学者的关注．近几十年来，这一领域的研究已取得了大量的成果．在本文的绪论中介绍了迭代和迭代方程的有关概念、迭代与动力系统的关系，并综述了近年来国内外数学家对迭代方程取得的成果。关于迭代的目前已知的结果大多是在确定性系统中获得的，在不确定性系统中的迭代研究是比较少的，特别是关于迭代方程在不确定性系统中的研究是屈指可数的，在确定性系统中，所涉及的函数通常具有较好的性质或特征，比如单调性、连续性、可以考虑导数等，但是在不确定性系统中的映射则由于更复杂而不具备这些性质或特征．因此讨论不确定系统的迭代需要在方法上有发展，在绪论中我们综合性地介绍了集值映射迭代方程和模糊动力系统的发展概况及一些未解决的问题． 确定性动力系统理论是建立在分明拓扑空间上的，历史悠久，自1965年以来，国内外许多学者致力于将分明拓扑推广至格上，已经形成系统的L-拓扑理论，第二章首先引入集值分析中的一些基本概念和结果，其次从完备格的基本概念开始，介绍了格值映射、L-拓扑的有关知识。更多还原

【Abstract】 The purpose of deterministic dynamical system theory is to study rules of change in state which depends on time. Usually there are two basic forms of dynamical systems: continuous dynamical systems described by differential equations and discrete dynamical systems described by iteration of mappings. Many mathematical models in physics, mechanics, biology and astronomy are given in such forms. Iterative equations are the equivalent form that includes iteration of unknown mappings. It has many important applications in the nature. For example, the Feigenbaum phenomena as investigating universality of period-doubling bifurcation cascade, invariant curves and manifolds of a differential equation, invariant tori and curves of Hamiltonian systems and the normal form problem can be reduced to the research of iterative equations. Iterative equation has become an important mathematics equation form which closely links differential equations, difference equations, integral equations and dynamical systems, and attracts attentions of many scholars. In recent decades, many results have been given in this field. In Chapter 1, concepts of iteration and iterative equation as well as relationship between iteration and dynamical systems are introduced. Many known results on iterative equations which are all given for deterministic systems, are summarized. In particular, less results are given for iterative equations with indeterminate systems. For indeterminate systems functions do not have so good properties (e.g., monotonicity, continuity, differentiability, etc) as for deterministic ones. Hence it needs to make some improvement in discussing iteration in indeterminate systems. In Chapter 1, we survey the development of set-valued iterative equations and fuzzy dynamical systems, and give some unsolved problems in these fields.更多还原