广义模糊动力系统的Martelli混沌性质研究

The basic goal of the theory of discrete dynamical systems is to understand the nature of all orbits. However, in many real problems such as biological species, demography, etc., it is not sufficient to know as are moved the points in the base space but it is necessary to know as are moved the subsets of the space, what carries us to the problem of analyzing the dynamics of set-valued discrete systems. Moreover, when the available dates are vague then we can consider the discrete fuzzy dynamical systems. A so-called generalized discrete fuzzy dynamical system consists of a fuzzy topological space and a fuzzy mapping. We propose to conduct a study to explore some Martelli’s chaotic properties of the generalized discrete fuzzy dynamical system. Some topics of interest include: (1) the relations between Martelli’s chaotic properties of the original and fuzzified dynamical systems, (2) Martelli’s chaos in generalized discrete fuzzy inverse limit dynamical system, and (3) dynamics of semi-continuous generalized discrete fuzzy dynamical system. Moreover, the relations between dynamical properties of Martelli’s chatoic system and Wiggin’s chatoic system are investigated. The objectives of the present project are to develop the theory of discrete fuzzy dynamical systems, provide theoretical foundation for controlling of chaotic fuzzy systems.

动力系统的主要任务之一是研究系统轨道的渐进性质。在诸如人口统计、生物种群等现实问题中，仅考虑系统中单点的轨迹是不够的，还需考察系统子集的运动状态，这就是集值离散系统。当集值系统受不确定性因素影响时，可用模糊集来处理分明子集所携带的信息，从而研究在扩张映射迭代下的模糊集的运动状态。该类由扩张映射与模糊拓扑空间构成的系统称为离散模糊动力系统，它由分明系统诱导而来。本项目主要研究由广义扩张映射与模糊拓扑空间所构成的广义模糊动力系统的Martelli混沌性质，包括：（1）离散系统与诱导广义模糊系统之间的Martelli混沌性质关系研究。（2）广义模糊逆极限动力系统的Martelli混沌性质研究。（3）半连续广义模糊动力系统的Martelli混沌性质研究，讨论Martelli与Wiggins混沌系统之间的内在联系。项目研究成果将丰富和发展模糊系统的动力学理论，为模糊系统的混沌控制提供理论支撑。

复杂系统往往伴随着模糊性，建立在结构或者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、心理学等系统的一个共同特点是具有多层次的混合结构，该类结构主要表现为离散时间的动力学， 离散系统的动力学研究已有相当积累，但尚不完善，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之间的动力学性质关系，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，主要研究了：（1）离散系统与诱导广义模糊系统之间的Martelli混沌性质关系，回答了Roman-Flores于2008年提出的公开问题；（2）模糊数商空间的拓扑与代数性质，得到一个完备模糊数商空间，并在此完备模糊数商空间（在新同余意义下）上引入模糊逆极限动力系统；（3）利用不动点理论在模糊Banach空间中讨论了模糊集值泛函方程的Hyers-Ulam-Rassias稳定性问题。希望通过研究模糊泛函方程的稳定性问题，进一步探讨系统轨道的渐进性质，包括Martelli混沌性质。上述结果为我们提供了从已知系统的动力性态了解未知系统的运动规律的研究思路与方法，丰富和发展了不确定性动力系统理论。