易經渾沌及渾沌同步與控制

Abstract

自Lorenz 在1963 年發現渾沌現象以來，已達四十六年。有關渾沌之理論、性質及特徵等之研究 及渾沌控制、渾沌化、渾沌同步等應用研究，汗牛充棟，不可勝數。但所有研究皆限於諸渾沌系統在 時間正向流逝( t → +∞ )中之研究。在真實物理中，時間知倒流迄今尚未發現。然而在數學中，為了研 究的完整性，時間倒流(t→ −∞)實屬必須研究之範圍。2009 年9 月，本人與李仕宇(本人之博士研究 生)首次發表時間倒流( t → −∞ )中的Lorenz 渾沌於期刊論文。個人認為此舉對渾沌研究具有劃時代之 意義。打一個也許不甚確切的比喻：哥倫布之前，大多數人類活動範圍皆集中於東半球。哥倫布之後， 人類活動範圍乃擴展至西半球，即擴展至整個地球矣。由是觀之，以往僅就t → +∞ 之極多渾沌研究 實屬片面，應補充t : 0 → −∞ 之研究，方為全面之研究。僅此一點即可產生難以計量之有價值之研究 論文。 再者，中國哲學之源頭是易經。易經不僅是五經之首，也是五經之始。易經之基本原理為太一出 兩儀(即陰陽)兩儀出四象，四象出八卦。三爻為一掛。八掛經三爻排列組成64 掛。易經的根本範疇是 陰陽。陰指負，柔，保守等；陽指正，剛，進取等。陰與陽相反相成，異中有同，同中有異。上述論 文已驗證了陰渾沌( t : 0 → −∞ )與陽渾沌( t : 0 → +∞ )的異中有同，同中有異之特徵。這可能也是陰陽學 說第一次在精確的科學中得到證實。本研究計畫嚴格限於易經之科學理論，決不涉及迷信附會。 當然，孤證不足行。本計劃之主要宗旨在於尋求多種之渾沌系統的陰渾沌之存在(這是很困難的， 只有系統參數湊的正好，才能找到陰陽渾沌)，及其陰、陽渾沌之異同，試圖找出其間的規律性(如果 有的話)。此其一。宗旨之二為應用陰渾沌研究渾沌同步及控制，及應用陰陽渾沌之搭配，研究以上諸 新渾沌系統之渾沌同步及控制。三爻為一掛及八掛皆可實現。六十四掛由八掛之排列組合皆可實見。 研究重點為： 1. 覓求Ro&&sller ，Chen Lu&& 及Chen-Li 諸自治系統太一渾沌 2. 覓求Ro&&sller 系統，Chen 系統， Lu&&系統及Chen-Li 諸系統之陰渾沌。比較其陰陽渾沌之異同，試 圖找出陰陽渾沌之間之規律特徵，例如Faigenbaum 常數。 3. 利用陰渾沌本身及陰陽渾沌之搭配得出以上諸新系統之渾沌同步，渾沌控制等。 4. 利用陰陽渾沌經排列組合搭配得以上四系統之八掛渾沌及其渾沌同步，渾沌控制，利用線性控制， 適應控制，強健控制等 5. 利用本八掛以外四系統之八掛渾沌及用排列組合成8×8=64 掛，完成易經之全部理論(工作量甚 大)，渾沌同步及控制採線性控制，適應控制，強健控制等。 6. 研究Duffing，van der Pol，Mathieu，Morse 諸非自治系統的以上一至五的五點內容。 7. 研究Ikeda，Ikeda-Lorenz，Ikeda-R&o& ssler，Ikeda-L &u& 諸時滯系統的一至五的五點內容。

Since Lorenz discovered chaos in 1963, 46 years passed. In these 46 years, uncountable papers studied the theories, properties and characteristics of chaos, chaos controls, chaos anticontrols and chaos synchronizations. However, all these studies were limited to the investigations of chaotic systems for positive time direction tion ( t → +∞ ). In real physical world, the backward time is still not discovered, but in mathematics, the backward study actually must be studied for completeness. In 2009, I and Shih-Yu Li (my doctoral graduate student) published a paper concerning forward and backward Lorenz chaos in a well-known journal firstly. This is epoch-making for chaos study, because for completeness of study, the uncountable studies for t → +∞ must be studied again for t → −∞ . Give a maybe unexact metaphor: Before Columbus, the activity of mankind concentrated in east hemisphere; after Columbus, the activity expanded to west hemisphere, i.e. to the whole globe. Therefore the study of chaos, for completeness, must concerns both t → +∞ , and t → −∞ . In the very past, the studies only for t → +∞ are all unilateral. The fundamental principle of Yi is that Great one borns two ultimates,two ultimates born four quadrants four quadrants light hexagrams, eight hexagrans born 8×8=64 hexagrams. In addition, the origin of Chinese Philosophy is the Yi classic which is the first and beginning of Chinese Five Classical Books. The fundamental categories of the Yi classic are Yin and Yang. Yin means negative, soft and conservative; Yang means positive, strong and aggressive. They are both contrary and complementary to each other. They appear both sameness with difference and difference with sameness. The above-mentional paper exactly verify these characteristics of Yin ( t → −∞ ) and Yang ( t → +∞ ). This may be the first time to verify the theory of Yin-Yang in exact science. The project is strictly limited to the scientific principle of Yi classic, does not concerns the superastitious interpretation. Of course, single verification is far from soundness of a statement. The main aim of this project is to find a lot of Yin chaos for chaotic systems, with the similar but different Yin-Yang chaos try to find certain regulation between them if possible and give the application of Yin chaos and Yin-Yang chaos with their combinations and permutations, for chaos synchronizations, chaos controls. The main parts of our study are: 1. Find The Yin chaos of four autonomous chaotic systems, i.e. Find great one chaos of Ro&&sller system, Chen system, Lu&& system and Chen-Li system. Try to find out the regulation between the Yin chaos and Yang chaos of these four new systems. 2. Find The Yin chaos of four autonomous chaotic systems, i.e. Ro&&sller system, Chen system, Lu&&system and Chen-Li system. Try to find out the regulation between the Yin chaos and Yang chaos of these four new systems. 3. Apply the Yin chaos and Yin chaos plus Yang chaos with their combination and permutation for chaos synchronizations chaos control and chaotizations. 4. By combination and permutation, find the eight hexagrams of Ro&&sller , Chen, Lu&&, Chen-Li systems. Synchronization and control are gaved by direct, adaptive, robust control etc. 5. By combination and permutation, find the sixty four hexagrams, complete fundamental principle of Yi classic Synchronization and control are accomplished by direct control, adaptive control and robust control.(The work is laborious) 6. Study the above 1-5 trems for nonautonomous systems, i.e. Duffing, van der Pol, Mathieu, Morse systems. 7. Study the above 1-5 trems for nonautonomous systems, i.e. Ikeda, Ikeda-Lorenz, Ikeda-R &o& ssler, Ikeda-L &u& systems.