藉由純誤差動態方程與精巧李亞普諾夫函數所達成之新渾沌系統廣義同步，非完整系統之渾沌現象，不同渾沌系統之變尺度時間非等時交織同步與不同渾沌系統之雙交織同步

Abstract

本論文探討藉由純誤差動態方程與精巧李亞普諾夫函數所達成之新渾沌系統廣義同步，非完整系統之渾沌現象，不同渾沌系統之變尺度時間非等時交織同步與不同渾沌系統之雙交織同步。 首先，提出以兩組不同的帶阻尼非線性Mathieu系統之相互線性耦合而構成的新渾沌系統，並研究其規律與渾沌動態行為。 其次，利用純誤差動態方程、精巧李亞普諾夫函數與精巧非對角化李亞普諾夫函數來達成廣義同步。採用不需要數值模擬輔助的純誤差動態方程來實現廣義同步，以取代目前常用的含主從狀態變量的混合誤差動態方程。此外還採用精巧李亞普諾夫函數與精巧非對角化李亞普諾夫函數，以取代目前廣泛使用，一成不變，且大幅地削弱李亞普諾夫直接法威力的平方和李亞普諾夫函數。在數值模擬中以新渾沌系統為範例。 接著，首次完整地確認了非完整系統的渾沌現象，包括具有外加非完整約束的非完整系統，如目標為直線振動的追蹤問題與目標為繞圓周旋轉的追蹤問題，及具有外加非線性非完整約束的非線性非完整系統，如速度大小保持不變的問題。渾沌的研究範圍首次被拓展至非完整系統與非線性非完整系統。系統的動態方程式是藉由基本非完整形式的拉格朗日方程與非線性非完整形式的拉格朗日方程而導出。透過所有的渾沌數值判據，包括最可靠的李雅普諾夫指數、相位圖、龐卡萊圖與分歧圖，首次完整地證明了渾沌現象存在於非完整系統與非線性非完整系統。並更進一步地發現費根堡常數法則在非線性非完整系統中依然適用。 此外，提出了一種新型的渾沌同步，稱為“非等時交織同步”，可表達為y(t)=F(x(tau),y(t),t)，其中tau是時間t之給定函數，即所謂的變尺度時間。它是廣義同步的延伸，之所以命名為“非等時交織同步”是因為y(t)扮演著交織的角色，且x(t)與y(t)分別在不同的時刻tau及t達成同步。當非等時交織同步應用在秘密通訊時，由於函數關係比傳統廣義同步的函數關係更為複雜，且對於攻擊者而言，除了破解系統的模型和複雜的函數關係，還多了破解變尺度時間tau的困難，因此傳送訊號時採用非等時交織同步會比採用傳統廣義同步更難被偵測到，可用來加強秘密通訊的安全。在此以非線性控制與適應控制的方法來實現非等時交織同步。使用適應控制時，估測到的Lipschitz 常數遠小於使用非線性控制所求得的Lipschitz 常數，這使得控制器的增益大幅減少，換句話說，控制器的成本隨之而降。採用的控制方法可有效地適用於自治與非自治渾沌系統，無論x(tau)與y(t)系統的維度相同與否。 並更進一步地提出一種新型的渾沌同步，稱為“雙交織同步”，可表達為G(x,y)=F(x,y,t)。它是交織同步y=F(x,y,t)的延伸，因為交織函數同時出現在等式的左右兩邊，故命名為“雙交織同步”。利用其同步形式的複雜性，可用來加強秘密通訊的安全。基於Barbalat引理，提出以主動控制達到雙交織同步的方法，並成功地應用在自治與非自治渾沌系統。

Generalized synchronization of new chaotic systems by pure error dynamics and elaborate Lyapunov function, chaos of nonholonomic systems, non-simultaneous symplectic synchronization of different chaotic systems with variable scale time, and double symplectic synchronization of different chaotic systems are studied in this thesis. Firstly, the new chaotic systems constructed by mutual linear coupling of two non-identical nonlinear damped Mathieu systems are introduced, and the regular and chaotic dynamics of the new chaotic systems are studied. Then, by applying pure error dynamics, elaborate Lyapunov function, and elaborate nondiagonal Lyapunov function, the generalized synchronization is obtained. Instead of current mixed error dynamics in which master state variables and slave state variables are presented, the generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation. The elaborate Lyapunov function and the elaborate nondiagonal Lyapunov function are applied rather than the current monotonous square sum Lyapunov function, deeply weakening the powerfulness of Lyapunov direct method. New chaotic systems are used as examples with numerical simulations. Chaos of nonholonomic systems with external nonholonomic constraint, the straightly oscillating target pursuit problem, or the circularly rotating target pursuit problem, and chaos of nonlinear nonholonomic system with external nonlinear nonholonomic constraint, the magnitude of velocity keeping constant, is completely identified for the first time. The scope of chaos study is firstly extended to nonholonomic systems and nonlinear nonholonomic system. By applying the fundamental nonholonomic form of Lagrange’s equations and the nonlinear nonholonomic form of Lagrange’s equations, the dynamic equations are expressed. The existence of chaos in nonholonomic systems and nonlinear nonholonomic systems are firstly completely identified by all numerical criteria of chaos, i.e. the most reliable Lyapunov exponents, phase portraits, Poincaré maps and bifurcation diagrams. Furthermore, it is found that the Feigenbaum number rule still holds for nonlinear nonholonomic system. We propose a new type of synchronization, “non-simultaneous symplectic synchronization”, y(t)=F(x(tau),y(t),t), where tau is a given function of time t, so-called variable scale time. It is an extension of generalized synchronization and it is called “non-simultaneous symplectic synchronization” due to y(t) plays the “interwined” role and the synchronization is achieved at “different time” for x(tau) and y(t). When applying the non-simultaneous symplectic synchronization in secret communication, since the functional relation of the non-simultaneous symplectic synchronization is more complex than that of the traditional generalized synchronization, and cracking the variable scale time tau is an extra task for the attackers in addition to cracking the system model and cracking the functional relation, the message is harder to be detected by applying the non-simultaneous symplectic synchronization than by applying traditional generalized synchronization. Therefore, the non-simultaneous symplectic synchronization may be applied to increase the security of secret communication. Nonlinear control and adaptive control are applied to obtain the non-simultaneous symplectic synchronization. The estimated Lipschitz constant obtained by applying adaptive control is much less than the Lipschitz constant obtained by applying nonlinear control. This result in the reduction of the gain of the controller, i.e. the cost of controller is reduced. The proposed scheme is effective and feasible for both autonomous and nonautonomous chaotic systems, whether the dimensions of x(tau) and y(t) systems are the same or not. Furthermore, a new type of synchronization, “double symplectic synchronization”, G(x,y)=F(x,y,t), is proposed in this thesis. It is an extension of symplectic synchronization, y=F(x,y,t). Since the symplectic functions are presented at both the right hand side and the left hand side of the equality, it is called “double symplectic synchronization”. The double symplectic synchronization may be applied to increase the security of secret communication due to the complexity of its synchronization form. By applying active control, the scheme of double symplectic synchronization is derived based on Barbalat’s lemma, and it is applied successfully to both autonomous and nonautonomous chaotic systems.