Qualitative Analysis of Complex Dynamical Systems

本项目将尝试建立几类用于联想的人工反馈神经网络的局部和全局定性结果回忆。这些结果将被系统地用于建立合成结果。P.I.将考虑连续时间和离散时间神经网络，这些神经网络可能具有线性或(高阶)非线性并且与神经元的通常的S型模型或更高阶的非线性模型(例如，楼梯非线性)。全局定性结果将解决状态空间中均衡的分布、全局稳定性等问题。局部定性结果将涉及平衡点吸引域的稳定性性质和估计，轨迹的收敛速度等，综合结果将解决学习和遗忘算法，存储容量，私家侦探还将利用这些神经网络结果来确定具有饱和非线性的一般动力系统(包括控制系统)的分析和设计技术。这样的系统在科学和技术中无处不在。除了局部和全局的定性表征外，我们还寻求可控性、可观性、稳定化等结果。在控制系统的情况下。

This project will attempt to establish local and global qualitative results for several classes of artificial feedback neural networks for associative memories. These results will be used in a systematic way to establish synthesis results. The P.I. will consider continuous-time and discrete-time neural networks which may be endowed with linear or (higher order) nonlinear interconnecting structures and with the usual sigmoidal models or higher order nonlinear models for the neurons (e.g., staircase nonlinearities). Global qualitative results will address the distribution of equilibria in the state space, global stability, and the like. Local qualitative results will address the stability properties and estimates of the domain of attraction of equilibria, rates of convergence of trajectories, and so forth, Synthesis results will address learning and forgetting algorithms, storage capacity, etc. The P.I. will also use these neural networks results to establish analysis and design techniques for general dynamical systems (including control systems) endowed with saturation nonlinearities. Such systems are pervasive in science and technology. In addition to local and global qualitative characterizations, we seek results for controllability, observability, stabilizability, etc., in the case of control system.