基于模糊集理论的离散系统动力学研究

Fuzziness is an inherent feature of complex systems. We need a theory to formulate human knowledge in a systematic manner and put it into engineering controls, biology, psychology, etc. The introduction of fuzzy set theory into the field of dynamical system is motivated by the needs for a theory of systems whose structure and/or behavior involve uncertainties, and the design of fuzzy controllers. The common feature of such complex systems is hierarchy of hybrid structure, which mainly described by dynamics of discrete systems. Discrete dynamical systems have been extensively studied. However, there remain some problems, especially the dynamics of nonautonomous systems. Moreover, the uncertainty precents new and different challenges for the research. It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e., a discrete dynamical system on the space of fuzzy sets. The discrete fuzzy dynamical systems can be generated by iteration of fuzzy mapping operators. We can understand dynamics of unkown systems by investigating the relations between the original and fuzzified systems. In this project we study relations between dynamical properties of the original and fuzzified dynamical systems. Several themes run through the present project, we show that: (1) the relations between different chaotic fuzzy systems. We try to find a reference system to describe others, which could simplify the research, (2) the relations between sensitivity dependence on initial conditions of the original and fuzzified dynamical systems to reveal the essence of chaos, and (3) the motion law of the nonautonomous dynamical systems by.introducing a new concept of nonautonomous fuzzy dynamical system. We shall apply some new methods to understand the aperiodicity of the orbit of nonautonomous fuzzy dynamical system. This part of the research is an innovation of this project.

复杂系统往往伴随不确定性，建立在结构或行为上具有不确定性的系统理论对研究复杂系统的控制问题具有重要价值。工程控制、生物学等系统的一个共同特点是具有多层次的混合结构，此类结构主要表现为离散时间的动力学。动力系统的离散化归结为映射的迭代。目前，离散系统中的非自治系统理论正蓬勃发展，复杂系统自身的不确定性又带来了新的挑战。通过探讨分明系统与诱导模糊系统的动力学之间的关系，能从已知系统的动力性态了解未知系统的运动规律，本项目以此为主线，拟研究：（1）不同混沌定义下模糊系统的关系。试图找到一类参照系统，讨论其与分明系统的关系，简化因混沌定义林立而造成研究分散的问题。（2）分明系统与诱导模糊系统的各类敏感初条件之间的关系，揭示混沌运动的本质特征。（3）引入非自治离散系统诱导的模糊系统，构造新工具来刻画系统轨道的非周期性，探寻非自治系统的运动规律，这在研究内容上是一种创新与探索。

复杂系统往往伴随不确定性，建立在结构或行为上具有不确定性的系统理论对研究复杂系统的控制问题具有重要价值。工程控制、生物学等系统的一个共同特点是具有多层次的混合结构，此类结构主要表现为离散时间的动力学。动力系统的离散化归结为映射的迭代。目前，离散系统中的非自治系统理论正蓬勃发展，复杂系统自身的不确定性又带来了新的挑战。通过探讨分明系统与诱导模糊系统的动力学之间的关系，能从已知系统的动力性态了解未知系统的运动规律，沿着这个方向，本项目研究了：.（1）非自治离散动力系统的弱稳定性。在非自治动力系统中引入“弱稳定性”定义，给出弱稳定点集的刻画，分析了弱稳定性与伪轨跟踪之间的关系，证明了分明非自治系统是弱稳定的当且仅当其诱导集值系统是弱稳定的。.（2）分明动力系统与诱导模糊系统各类敏感初条件之间的关系。分别探讨了了非自治系统与诱导模糊系统的敏感性、强敏感性与中值敏感性之间的关系，并以反例说明对于敏感性而言，两系统之间的不等价性。.（3）分明动力系统与诱导模糊系统伪轨跟踪性之间的关系。在模糊系统中引入一个比伪轨跟踪强的性质—所谓的链连续定义，探讨了分明离散动力系统与诱导模糊系统在链连续性之间的联系。.项目通过对分明系统与诱导模糊系统动力学行为之间的关系研究，发掘分明系统与诱导模糊系统在动力学行为之间的联系，丰富了模糊动力系统的研究成果，达到利用已知系统刻画未知系统的目的。