Study on Structures and Dynamics of Nonlinear Systems

随着超大规模集成电路的迅速发展和孤子、混沌等新的非线性现象的发现，对非线性动力学系统的分析变得非常重要。本课题的目的是通过研究人员发展的新的数学方法，阐明非线性系统结构及其动力学的基本性质，并为其在实际问题中的应用奠定基础。本课题已完成三年多，其富有成果的成果是：1.利用原有的非确定性算子理论，建立并相互重组了下列理论：1.非线性系统的内部结构和参数的总体相关性的评价理论。用原有的基于同伦或分解的方法进行非线性系统数值分析的理论。分析孤子、混沌等非线性现象的新方法。此外，利用上述理论，我们得到了以下结论：4.从一个极其简单的电路中观察到一个非周期吸引子，并严格证明了该吸引子是混沌的。并联风机和多体系统的建模及其动力学分析。分析了光传输和生物系统的模型。

With the rapid advance of VLSI and the discovery of new nonlinear phenomena such as soliton and chaos, analysis of nonlinear dynamic systems becomes very important. The purpose of this project is to clarify the fundamental properties of both the nonlinear system structure and its dynamics, and to buid the foundation of their applications to the practical problems, by means of the new mathematical methods which the present researchers have developed.This project has been accomplished over three years, and its fruitful result is establishment and mutual recomposition of the following theories :1. Theory on evaluation of the overall dependence on internal structures and parameters of nonlinear systems, by making use of the original theory of nondeterministic operators.2. Theory on numerical analysis of nonlinear systems by the original methods based on homotopy or decomposition.3. New approach for the analysis of nonlinear phenomena such as soliton and chaos. Furthermore, using the above theories, we threw light on the followings :4. Observation of a non-periodic attractor from an extremely simple circuit, and the rigorous proof of the attractor to be chaotic.5. Modelling of parallel blower and multibody systems and analysis of their dynamics.6. Analysis of the models of optical transmission and biological systems.