Qualitative Analysis of Complex Dynamical Systems

研究了复杂动力系统的定性性质。这样的系统可以由相互关联的大型微分方程组、积分方程组、偏微分方程组或这些类型的方程的组合在数学上表示。这项工作将研究振荡的存在和稳定性，平衡点的稳定性，以及这类系统的输入输出性质。这些课题显然在理论研究和具体技术问题的应用中都具有重要意义。这些方法将涉及泛函分析技术、变换、积分流形理论的结果、微分方程组和积分方程组的比较理论以及积分方程组的预解算子理论。期望这一研究结果将为研究复杂反馈系统的振荡行为提供新的方法。它们还将为分析大型复杂动力系统反馈控制系统的稳定性提供新的方法。

The qualitative properties of complex dynamical systems are investigated. Such systems could be mathematically represented by large interconnected systems of differential equations, integral equations, partial differential equations, or a combination of these types of equations. The work will investigate the existence and stability of oscillations, the stability of the equilibria, and the input-output properties of such systems. These topics are clearly of great importance in theoretical studies and in application to specific technological problems. The methods will involve functional analysis techniques, transformations, results from the theory of integral manifolds, comparison theory for differential and integral equations and the theory of resolvent operators for integral equations. It is expected that the results of this research will provide new methods to investigate oscillatory behavior of complex feedback systems. They will also provide new methods of analyzing stability properties of feedback control systems for large and complex dynamical systems.