A Fundamental Study on Dynamics and Control of Complex Systems

本研究作为大气、生命、经济等复杂系统动力学与控制的基础性研究，建立了耦合振子非线性行为分析技术和混沌控制方法，主要内容如下：1.对于一般的两自由度周期强迫耗散系统，提出了一种分析共振周期轨道的同宿和异宿行为的理论方法，并描述了混沌的产生机制。自由哈密顿系统的鞍中心，理论技术，分析同宿和异宿行为的周期轨道和不变环面附近的鞍中心。应用这些技术，我们描述了一个无限自由度模型中的复杂行为的无强迫和无阻尼，屈曲梁。3.一个软件包的计算机代数系统，Mathematica，实现必要的计算高阶平均法，和一个驱动程序的计算机软件名为numerical的数值分析的同宿和异宿的行为。对受迫单摆和耦合摆的非线性行为进行了理论分析，并通过数值模拟和实验验证了理论结果。此外，一个两连杆机器人机械手理论和数值证明表现出复杂的behavior5。应用改进的延迟反馈和外反馈方法，对受迫单摆和耦合摆进行了混沌控制，并通过数值模拟和实验验证了其在噪声影响下的有效性。

In this research, as a fundamental study on dynamics and control of complex systems like atmosphere, life and economic systems, analytical techniques for nonlinear behavior and chaos control methods in coupled oscillators were established as follows.1.For general two-degree-of-freedom dissipative systems with periodic forcing, a theoretical technique to analyze homoclinic and heteroclinic behavior for resonant periodic orbits was proposed and a mechanism for chaos was described.2.For multi-degree-of-freedom Hamiltonian systems with saddle-centers, theoretical techniques to analyze homoclinic and heteroclinic behavior for periodic orbits and invariant tori near the saddle-centers were developed. Applying these techniques, we describe complicated behavior in an infinite-degree-of-freedom model for an unforced and undamped, buckled beam.3.A package of the computer algebra system, Mathematica, to implement necessary computations for the higher-order averaging method, and a driver to a computer software called AUTO for numerical analysis of homoclinic and heteroclinic behavior were developed. Their usefulness was demonstrated for several examples.4.Nonlinear behavior in a forced, simple pendulum and coupled pendula was analyzed theoretically, and the theoretical results were demonstrated in numerical simulations and experiments. Moreover, a two-link robot manipulator was theoretically and numerically proven to exhibit complicated behavior5. Applying improved versions of delayed and external feedback, we performed chaos control in a forced, simple pendulum and coupled pendula, and demonstrated their effectiveness under an influence of noise in numerical simulations and experiments.