【摘要】：一直以來,混沌(Chaos)都是非線性科學(xué)研究的熱點問題之一,而奇異吸引子則是反映系統(tǒng)混沌運動的典型特征,故而對奇異吸引子的產(chǎn)生機制,存在性條件,以及吸引子本身性質(zhì)的探討有著重要的意義。自從1963年,著名的Lorenz方程在數(shù)值求解被發(fā)現(xiàn)存在吸引子到1999年,首次利用規(guī)范形并結(jié)合計算機輔助證明Lorenz方程的確存在奇異吸引子,人們對奇異吸引子的分析和研究已逐步深入。其中,研究最多的是為模擬Lorenz方程的動力學(xué)行為而提出的幾何Lorenz吸引子的模型。向量場的分支理論主要研究動力系統(tǒng)的軌道族的拓?fù)浣Y(jié)構(gòu)隨參數(shù)變化所發(fā)生的變化及其變化規(guī)律。當(dāng)存在鞍點的同宿軌時,系統(tǒng)是結(jié)構(gòu)不穩(wěn)定的,因此,同宿軌分支蘊含著豐富的動力學(xué)行為。在Lorenz模型中,對應(yīng)于蝴蝶同宿的分支,兩個對稱的同宿軌道破裂時,每個回路產(chǎn)生一個周期軌道,并且這兩個周期軌道的穩(wěn)定流形與不穩(wěn)定流形彼此橫截相交。這種橫截同宿現(xiàn)象導(dǎo)致了眾多的復(fù)雜現(xiàn)象。本文對可能產(chǎn)生洛倫茲型吸引子的兩種余維2的對稱同宿軌分支：傾斜翻轉(zhuǎn)分支和軌道翻轉(zhuǎn)分支,分別進行了詳細(xì)的研究分析。具體的,文中以含有鞍點的3維Cr(r≥3)對稱系統(tǒng)X=f(X),X∈R3為研究對象,首先在鞍點平衡態(tài)附近將系統(tǒng)化為簡單的、便于分析的形式,由此構(gòu)造Poincare返回映射。根據(jù)得到的Poincare返回映射對傾斜翻轉(zhuǎn)和軌道翻轉(zhuǎn)情形的分支情況進行了討論,并得到分支曲線圖。隨后討論了這兩種分支情形的開折中洛倫茲型吸引子的存在性問題。通過對幾何洛倫茲模型的分析,以及一維洛倫茲映射的拓?fù)湫再|(zhì)的討論,并將二維Poincare返回映射約化到一維,最終得到了洛倫茲型吸引子的存在范圍。
[Abstract]:Chaotic (Chaos) has always been one of the hot issues in nonlinear science, and singular attractor is a typical feature of chaotic motion of the system. It is of great significance to discuss the properties of the attractor itself. Since 1963, when the famous Lorenz equation was numerically solved, the existence of attractors was discovered. In 1999, the existence of singular attractors in Lorenz equation was proved for the first time by using canonical form and computer aid. The analysis and study of strange attractors have been gradually deepened. Among them, the geometric Lorenz attractor model proposed to simulate the dynamic behavior of Lorenz equation is the most studied. The bifurcation theory of vector field is mainly concerned with the variation of the topological structure of the orbital family of the dynamical system with the change of the parameters. When there is a saddle point homoclinic orbit, the system is structurally unstable, so the homoclinic orbit bifurcation contains abundant dynamic behavior. In the Lorenz model, when two symmetric homoclinic orbits break up, each loop produces a periodic orbit, and the stable manifold of the two periodic orbits intersects with the unstable manifold. This transversal homoclinic phenomenon leads to many complicated phenomena. In this paper, the symmetric homoclinic bifurcation of two kinds of codimension 2 which may produce Lorentz attractor is studied and analyzed in detail, that is, tilting overturning branch and orbit overturning branch. Specifically, in this paper, we take the 3-dimensional C r (r 鈮，

本文編號：2198498