新渾沌系統與新渾沌控制及同步方法

Abstract

渾沌系統之研究在物理、化學、生理學、各種工程等方面皆有日益重要之廣泛應用。Duffing系統、van der Pol系統、非線性Mathieu系統都是最重要的典型的渾沌系統。Ikeda系統及Mackey-Glass系統則是最重要的典型的時滯系統。兩種慣性測速器新系統也是重要的機械系統。本計畫採取適當的耦合方式構成新創的Duffing-van der Pol系統、Mathieu-Duffing系統、Mathieu-van der Pol系統、Ikeda- Mackey-Glass系統，從而擴大了各原來單純係統的研究範圍也深化了研究內容。兩種慣性測速器系統也是創新的，有實用價值的機械系統，值得研究。 渾沌控制與同步在物理系統、化學系統、生物系統、各種工程系統、秘密通訊、神經網路、自組織等方面的廣泛應用一日千里。本計畫提出一種新渾沌控制反控制方法、兩種新渾沌同步方法，具有重要的理論與實際意義：一、不同系統實用渾沌適應控制反控制新方法。傳統的渾沌控制侷限於將同一系統之渾沌運動控制至同一系統之週期解或平衡點。此法不僅可將本系統之渾沌控制到另一系統之週期解或平衡點，也可將本系統之週期解或混沌解反控制到另一更複雜系統之渾沌運動。而且涉及的各系統參數可以都是未知參數。大大地擴展了渾沌控制反控制的能力。二、非耦合渾沌同步新方法。目前之文獻絕大部分渾沌同步皆採耦合同步。以秘密通訊而言，耦合所需的主從系統狀態變量之傳送會造成失密及時滯之不良後果。非耦合同步可消除這些缺點。目前國際文獻多採以第三系統之渾沌變量或噪音對主從系統參數作同時激勵而獲同步。本計劃採（a）多管道下之各種形式之激勵（各種時間週期函數、渾沌函數、各種噪音）以保證同步之可靠性。（b）綜合性激勵（週期調製的渾沌函數、渾沌調製之噪音等）。（c）（a）、（b）同時進行以增加保密程度。（d）對分數階系統及時滯系統進行（a）、（b）、（c）三種激勵。三、指數後步渾沌同步新方法。目前渾沌同步皆取李氏函數為V正定、V負定，同步達成之時間較長控制品質不夠好。現採指數漸進穩定理論研究誤差系統零解，使同步完成時間大為減小。再結合後步(backstepping)設計可得三優點1.同步完成時間大減2.逐步選擇V函數，減少選V函數之難度3.減少控制器的數目。 & 研究重點為 1.Mathieu-van der Pol系統與Ikeda- Mackey-Glass系統的渾沌研究。用相圖、分歧圖、功率譜圖、李雅普諾夫指數分析渾沌之行為。 2.新式非耦合渾沌同步新方法。多管道、綜合性、分數階系統及時滯系統渾沌同步研究。 3.Duffing-van der Pol系統與Mathieu-Duffing系統之渾沌研究。用相圖、分歧圖、功率譜圖、李氏指數分析渾沌之行為，包括奇異吸引子之範圍及形狀、超渾沌之行為等。 4.實用渾沌適應控制反控制新方法。將原渾沌系統控制到不同系統之週期解或平衡點，此為渾沌控制。將原系統之週期解反控制為不同系統之渾沌解，或原系統之渾沌解反控制為不同系統更複雜渾沌解。此為混沌反控制。這些系統之參數一般皆未知，所以再加上適應控制法以達同步，再引用具概率觀念之實用穩定理論嚴格證明參數估計值趨近於未知值。 5.兩種慣性測速器新系統的渾沌行為研究，用相圖、分歧圖、功率譜圖、李雅普諾夫指數分析渾沌之行為。 6.指數渾沌同步新方法。使同步完成時間大為減少，結合後步（backstepping）設計使函數易於選擇，減少控制器數目。

Chaos systems have obtained wide applications in physics, chemistry, physiology, biology and various engineerings. Duffing system, van der Pol system and nonlinear Mothieu system all are paradigmatic chaotic systems in chaotic dynamics. Ikeda system and Mackey-Glass system are paradigmatic electro-optical and physiological system. Two kinds of inertial tachometer are also important mechanical systems. In this project, by suitable coupling, four new systems, namely, Duffing-van der Pol system, Mathieu-Duffing system, Mathieu-van der Pol systems and Ikeda-Mackey-Glass system are given. Two kinds of inertial tachometer are also new important systems. Chaos control and synchronization have rapidly extended their application for physical, chemical, biological systems, secure communication, neural networks, self-organization etc. In this project, a new chaos control and anticontrol method, two new chaos synchronization methods are proposed. They all deserve significant both theoretical and practical importance:1.pragmatical adaptive chaos control and anticontrol for different systems. Traditional chaos control and anticontrol only work for the same system. The new method extends the chaos control and anticontrol to other different systems, greatly increases its effectiveness. 2.new chaos synchronization method for uncoupled systems. For traditional synchronization by coupling, there exist defections of losing secret and lagging of signals, which can by eliminated by uncoupled synchronization. Traditional uncoupled synchronization are obtained by exciting two corresponding parameters of the systems to be synchronized by the same chaotic or noise signal. In this project, (a)multichanneled various excitations(various time function, chaotic function, noise, etc)are used to increase the reliability of synchronization in the accident of interruption of part of the channels.(b)Synthetic excitations(e.g. periodically modulated chaotic signal, chaos modulated noise, etc)are used to ensure the security. (c)(a)(b)are used simultaneously to ensure more security. (d)(a),(b),(c) are used for fractional order systems and time-delay systems. 3.New exponential backstepping synchronization method. In traditional chaos synchronization method, Lyapunov function is positive definite, is negative definite, the settling time of synchronization is rather long, the control quality is not satisfactory. In this project, exponential synchronizations used to increase the control quality greatly. Combined with backstepping design, three advantages are obtained: 1. Settling time is decreased greatly. 2.Vfunctions are chosen step by step, which becomes more easy. 3.The number of controller is decreased. The main parts of our study are: VV 1. The study of chaos of Mathieu-van der Pol system and Ikeda- Mackey-Glass system: By phase portraits, bifurcation diagrams, power spectra, Lyapunov exponents, the various chaotic behaviors of these systems will be studied. 2. New uncoupled synchronization method. Multichanneledly synthetically excited synchronizations for integral and fractional ordered, time-delay systems are studied. 3. The study of chaos of Duffing-van der Pol system and Mathieu-Duffing system: By phase portraits, bifurcation diagrams, power spectra, Lyapunov exponents, the various chaotic behaviors of these systems will be studied. The regions and shapes of the strange attractors, hyperchaos, ect will also be studied. 4. New pragmatical adaptive chaos control and anticontrol method for different systems. pragmatical asymptotical stability theory by probability concept is used to prove the estimated parameters must approach the unknown parameters. 5. The study of chaos of two kinds of inertial tachometer:By phase portraits, bifurcation diagrams, power spectra, Lyapunov exponents, the various chaotic behaviors of these systems will be studied. 6.New exponential backstepping synchronization method.