具延滯時間之藕合神經網路的動態

Abstract

本論文發展了一個新的方法學去探討具時間延遲之藕合神經網路系統的全局與局部動態行為。特別地，我們將這個方法實際運用到幾個神經網路模型上。其中所討論的全局動態行為包括了系統的全局同步化、抗同步化與全局收斂性。藉著觀察一些一維非自治方程式的幾何意義，並利用迭代推論來控制這些方程式；配合進一步的迭代討論進而確立以上提到的那些全局動態行為。所開發的方法可以同時用來推導出與延遲時間有關也可無關的結果。論文中所研究的局部動態行為包括了平衡點的穩定性和Hopf-分岔分析。我們的研究建立了一套異於線性穩定方法與李雅普諾夫函數方法的非典型方法來分析平衡點的穩定性；並且也可研究平穩點的吸引盆。通過觀察線性系統之特徵方程式的幾何結構並搭配Hopf-分岔理論，可推導出具體且容易驗證的條件來確保時間延遲所導致之同步週期解或異步週期解的存在性。 我們選取了三種類型的神經網路模型來實際應用這種方法學並以處理一些現有文獻中受到關心的議題。第一個模型是一般形式的Hopfield-type 神經網路。我們確立此類神經網路在擁有 個平衡點時的全局收斂性與平衡點的穩定性。我們的方法建立了一套有系統的方式來解決具相加形式之神經網路的收斂性與平衡點穩定性問題。而其餘兩個模型分別具最鄰近藕合形式的神經網路與具兩個環狀結構子網路的藕合神經網路。對於這樣的兩個模型，我們研究了系統的全局收斂性、全局同步化、平穩點的穩定性與延遲時間誘發的振盪與非同步。其所得到的結果可以映證或呼應一些現有文獻的數值模擬結果或推測。我們也在文中描敘幾個數值模擬，以佐證所獲得之理論。

We are interested in the collective dynamics for coupled systems with delays. To study global and local dynamics for some coupled systems, a new methodology is developed in this thesis. In particular, we implement this approach in several neural networks. Herein the global dynamics include global synchronization, anti-phase motion, global convergence (to single equilibrium or multiple equilibria) of the networks. Unfolded from the geometric structures of several associated non-autonomous scalar equations, an iteration scheme is designed to control the dynamics of these equations. Further iteration arguments are then developed to establish previous mentioned global dynamics for the networks. The approach we develop can be used to derive both delay-dependent and delay-independent criteria.The local dynamics we investigate include stability of the equilibria and Hopf bifurcation. Our studies establish the stability of equilibria by a nonstandard approach, as compared to the linearization with computation of the characteristic roots, and the Lyapunov function approach. Moreover, the basins of attraction for the stable quilibria can be investigated.Via a geometrical observation on the characteristic equation of the linearized system, the delayed Hopf bifurcation theory is adopted to guarantee the existence of delay-induced synchronous or asynchronous oscillations. The present approach is general and can be applied to several neural network models. To respond to some research issues in the literature, we investigate three neural network models.The first model is a general Hopfield-type neural network with delays. We establish the convergent dynamics and stability of the equilibria for such a networks with $3^n$ equilibria. Our study provides a systematic approach to investigate multistability and convergence of dynamics for additive-type neural networks. The latter two ones are the network with nearest-neighbor coupling and the network comprising two sub-networks with loop structure. For such two models, we investigate the synchronization, delay-induced oscillations and delay-induced asynchrony, the convergence of dynamics and stability of equilibria. Moreover, our results for the latter two models provide theoretical support to some numerical findings, and answer or respond to some conjectures in the existing literature. A number of numerical simulations are presented to illustrate our theory.