這篇論文分成三個部份。第一個部份研究 Transiently Chaotic Neural Network (TCNN) 系統中 Transversal Homoclinic Orbit 的存在性。第二個部份利用 Lyapunov function 來研究 TCNN 的穩定性。最後一個是研究 Discrete-Time Cellular Neural Networks (DT-CNN) 的混沌現象和穩定行為。這些定性的分析與研究有助於了解各別系統可能發生的行為。

My dissertation contains three parts. The subtitle of Part I is
''Transversal Homoclinic Orbits in a Transiently Chaotic Neural Network". Transiently chaotic neural network (TCNN) was proposed by Chen and Aihara~\cite{Aihara&Chen1995Chaotic}. We prove the existence of snap-back repellers in some parameters for TCNN. And, we generalize the result on the existence of
a Lyapunov function for TCNN with the constant self-feedback connection weight from symmetric connection weights to cycle-symmetric ones. The Part II is entitled ''Asymptotic Behaviors in a Transiently Chaotic Neural Network". We prove an extended version of LaSalle''s invariance principle for non-autonomous difference equations. Then, we apply the LaSalle''s invariance principle to TCNN with cycle symmetric connection. The subtitle of Part III is ''Dynamics for Discrete-Time Cellular Neural Networks".

Abstract i
1 Introduction...........................................1
1.1 Motivation.............................................1
1.2 Preliminary............................................2
2 Transversal Homoclinic Orbits in a Transiently Chaotic Neural Network.............................................6
2.1 Introduction ..........................................6
2.2 Illustrations of Snap-back Repellers for TCNN..........9
2.3 Basic Lemmas and the One-dimensional TCNN.............14
2.4 Snap-Back Repellers for TCNN..........................23
2.5 Asymptotic Convergence for TCNN.......................29
2.6 Numerical Illustrations...............................32
2.7 Conclusions...........................................35
2.8 Appendices............................................35
3 Asymptotic Behaviors in a Transiently Chaotic Neural Network...................................................39
3.1 Introduction..........................................39
3.2 LaSalle''s Invariance Principle for Non-autonomous Systems...................................................41
3.3 Lyapunov Function for TCNN............................44
3.4 Asymptotic Behaviors for TCNN.........................49
3.5 TCNN with Cyclic Updating Mode........................52
3.6 Conclusions...........................................56
4 Dynamics for Discrete-Time Cellular Neural Networks...57
4.1 Introduction..........................................57
4.2 Dynamics of the DT-CNN................................59
4.2.1 Complete stability..................................60
4.2.2 Illustrations of chaotic dynamics...................65
4.3 Verifications for the Dynamics of DT-CNN..............69
4.3.1 Complete Stability..................................69
4.3.2 Chaotic Behaviors...................................72
4.3.3 Numerical illustrations.............................80
4.4 Local and Global Saturated Patterns of DT-CNN.........81

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