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。对于具有周期强迫的一般二自由度耗散系统，提出了一种分析共振周期轨道同斜和异斜行为的理论方法，并描述了混沌的产生机理。对于具有鞍心的多自由度哈密顿系统，提出了周期轨道和鞍心附近不变环面同斜和异斜行为分析的理论方法。应用这些技术，我们描述了一个复杂的行为，在一个无限自由度的模型，无强迫和无阻尼，屈曲梁。开发了一个计算机代数系统软件包Mathematica，用于实现高阶平均方法的必要计算，并开发了一个名为AUTO的计算机软件的驱动程序，用于对同斜和异斜行为进行数值分析。通过几个例子证明了它们的有用性。对受迫摆、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，从理论上和数值上证明了双连杆机器人操纵臂具有复杂的行为。应用改进后的延迟反馈和外部反馈对强制摆、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它们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.