延遲型方程的穩定性與分歧性分析

Abstract

在生物、物理、工程系統中，延遲為一自然現象，故近年來仍持續地廣泛地受到重視。我們可從物理與工程等具代表性的期刊找出許多篇這方面的論文。另一方面，延遲型微分方程的數學研究至少已有40年的歷史。延遲微分方程的基本數學理論相當具有挑戰性，但也極富趣味。如延遲時間量與狀態有關之系統中，其解的存在性是最近才被研究出來的。基於過去幾年對類神經網路的研究經驗與興趣，我們計劃探討延遲性微分方程，與延遲性類神經網路，包括Hopfield neural networks with delays and cellular neural networks with delays。 我們有興趣的問題包括多平衡解(常態解)的存在性與穩定性分析、basins of attraction之估計、global behaviors、週期解-近週期解 (almost periodic)的存在性與穩定性、包含Hopf bifurcation 等之分歧現象、convergence of dynamics。各類型式的穩定性分析理論，包括linearization and characteristic roots，Lyapunov functions，Lyapunov functionals，direct estimations，oscillation analysis，也將是我們想要研究與發展的課題。以上若幹部份我們已得到一些成果。基於我們已有的研究成果與經驗，應該有相當不錯的機會在這些課題上有所進展。

There have been extensive scientific investigations in many physical, biological, and engineering systems with delays, due to that delay is a natural factor in these systems. On the other hand, fundamental mathematical theory for delayed equations, with more than forty years』 history, are interesting, yet challenging. For example, recently, the existence of solution for system with state dependent delay has just been established by Hans Otto Walther. In this project, we plan to study delayed differential equations and delayed neural networks including Hopfield neural networks with delays and cellular neural networks with delays. The problems we are interested in include existence of equilibrium and multiple equilibria, stability analysis, estimations of basins of attraction, global behaviors, periodic solutions, almost periodic solutions, Hopf bifurcations and other bifurcations, convergence of dynamics, monotone dynamics theory. We shall also focus on the theory of stability analysis, including linearization and characteristic roots, Lyapunov functions/Lyapunov functionals, direct estimations and Halanay inequality, oscillation analysis. Neural network models will serve as our trial and probing equations. Substantial contributions to the fields in the theory or applications of delayed systems will be expected in this project, based on the treatments we have developed in neural network models.