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与混沌控制的分析方法如下：1.对于一般具有周期强迫的两自由度耗散系统，提出了分析共振周期轨道同宿与异宿行为的理论方法，并描述了混沌产生的机理。2.对于具有鞍心的多自由度哈密顿系统，发展了周期轨道及鞍心附近不变环面的同宿与异宿行为的理论分析方法。应用这些技术，我们描述了无限自由度模型中无力和无阻尼屈曲梁的复杂行为。3.开发了用于实现高阶平均法的必要计算的计算机代数系统包数学，以及用于同宿和异宿行为数值分析的计算机软件AUTO的驱动程序。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对理论结果进行了验证。此外，还从理论和数值上证明了两连杆机器人机械手表现出复杂的行为。应用改进的延迟反馈和外反馈，对受迫单摆和耦合摆进行了混沌控制，并在数值仿真和实验中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.