本文關(guān)鍵詞： 混沌 奇異環(huán) 分段仿射系統(tǒng) 拓?fù)漶R蹄 龐加萊映射　出處：《華中科技大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：混沌現(xiàn)象是自然科學(xué)中廣泛存在但卻又十分有趣的動力學(xué)現(xiàn)象,在光滑動力系統(tǒng)中著名的Shilnikov類型的定理針對混沌不變集的存在性給出了嚴(yán)格的理論,這些定理部分地被推廣到分段光滑系統(tǒng)中。但是Shilnikov類型的定理中有一個非常重要的假設(shè)條件,即同宿軌或異宿環(huán)的存在性。對于一般的系統(tǒng)來說,探索系統(tǒng)同宿軌或異宿環(huán)的存在性是非常棘手的。幸運的是,針對分段仿射系統(tǒng)來說,我們不僅能夠顯式的表示各個子系統(tǒng)的穩(wěn)定流形和不穩(wěn)定流形,還可以顯式的表示各個子系統(tǒng)的解,因此分段仿射系統(tǒng)對于研究同宿軌或異宿環(huán)的存在性提供了良好的模型。在此基礎(chǔ)上還可以討論混沌不變集的存在性。本文正是致力于分段仿射系統(tǒng)同宿軌或異宿環(huán)的存在性以及混沌的研究,取得了如下創(chuàng)新成果：(1)三維分段仿射系統(tǒng)同宿軌存在性。研究了一類三維分段仿射系統(tǒng)的同宿軌的存在性,給出了與切換面橫截相交于兩點的同宿軌存在的充要條件,給出了構(gòu)造混沌系統(tǒng)的嚴(yán)格的數(shù)學(xué)方法。(2)三維分段仿射系統(tǒng)異宿環(huán)存在性及混沌。研究了一類三維分段仿射系統(tǒng)異宿環(huán)的存在性,給出了與切換面橫截相交于兩點的異宿環(huán)存在的充要條件,并在此基礎(chǔ)上運用拓?fù)漶R蹄理論給出了混沌不變集存在的嚴(yán)格證明。給出了構(gòu)造混沌系統(tǒng)的嚴(yán)格的數(shù)學(xué)方法。(3)四維分段仿射系統(tǒng)雙焦點同宿軌的存在性。研究了具有兩個子系統(tǒng)的四維分段仿射系統(tǒng)雙焦點同宿軌的存在性,給出了與切換面橫截相交于兩點的雙焦點同宿軌存在的充要條件,并給出了構(gòu)造混沌系統(tǒng)的嚴(yán)格數(shù)學(xué)方法。(4)四維分段仿射系統(tǒng)雙焦點異宿環(huán)的存在性。研究了具有兩個子系統(tǒng)的四維分段仿射系統(tǒng)雙焦點異宿環(huán)的存在性,給出了與切換面橫截相較于兩點的雙焦點異宿環(huán)存在的充要條件。并在此基礎(chǔ)上,構(gòu)造了一個具有雙焦點異宿環(huán)的四維系統(tǒng),給出了混沌不變集存在的計算機仿真結(jié)果。本文的具體內(nèi)容安排如下：第一章主要介紹了分段光滑系統(tǒng)的一些基本概念和分段光滑動力系統(tǒng)的研究現(xiàn)狀。第二章主要介紹了符號動力系統(tǒng)與拓?fù)漶R蹄理論。第三章主要研究了一類三維分段仿射系統(tǒng)同宿軌存在的充要條件,給出了一種構(gòu)造混沌系統(tǒng)的數(shù)學(xué)方法,并在此基礎(chǔ)上,構(gòu)造了幾個混沌系統(tǒng),給出了相關(guān)的計算機仿真結(jié)果。第四章介紹了一類三維分段仿射系統(tǒng)異宿環(huán)存在的充要條件,并在此基礎(chǔ)上運用拓?fù)漶R蹄理論證明了混沌不變集的存在性,給出了一種構(gòu)造混沌系統(tǒng)的數(shù)學(xué)方法。并在此基礎(chǔ)上,構(gòu)造了幾個混沌系統(tǒng),給出了相關(guān)的計算機仿真結(jié)果。第五章是四維分段仿射系統(tǒng)奇異壞的存在性。首先研究了一類四維分段仿射系統(tǒng)雙焦點同宿軌存在的充要條件,給出了一種構(gòu)造混沌系統(tǒng)的數(shù)學(xué)方法,并在此基礎(chǔ)上,構(gòu)造了一個混沌系統(tǒng),給出了相關(guān)的計算機仿真結(jié)果。其次研究了一類四維分段仿射系統(tǒng)雙焦點異宿環(huán)存在的充要條件,構(gòu)造了一個具有雙焦點異宿環(huán)的四維系統(tǒng),給出了混沌不變集存在的計算機仿真結(jié)果。第六章對全文的工作進(jìn)行了總結(jié),并對下一步工作擬定計劃。
[Abstract]:Chaos is a dynamic phenomenon but very interesting widely exists in the natural sciences in smooth dynamical systems in the famous Shilnikov type theorem for the existence of chaotic invariant set gives the strict theory, these theorems is partially extended to piecewise smooth system. But the Shilnikov type theorem is a very important the assumption that the existence of homoclinic or heteroclinic loop. For the general systems, explore the existing system of homoclinic or heteroclinic loop is very difficult. Fortunately, the piecewise affine systems, we can not only show that the stable manifold of subsystems and unstable manifold type that can also shows the various subsystems of the solution, so the piecewise affine system for the study of homoclinic or heteroclinic existence provides a good model. On this basis can be discussed. The existence of chaotic invariant set. This paper is devoted to study the existence of chaos and the piecewise affine system homoclinic or heteroclinic loop, the main contributions are as follows: (1) three dimensional piecewise affine system existence of homoclinic orbits. Existence of a class of three-dimensional piecewise affine systems of homoclinic orbits, are given with the switching surface transverse intersection in two necessary and sufficient conditions for existence of homoclinic orbits, a strict mathematical method for constructing chaotic system is presented in this paper. (2) three dimensional piecewise affine systems and existence of heteroclinic chaos. To study a class of three-dimensional piecewise affine systems heteroclinic the existence of heteroclinic loop gives necessary and sufficient conditions with the switching surface transverse intersection at two points exist, and based on the use of topological horseshoe theory gives the chaotic invariant sets are proved strictly. The strict mathematical method of constructing chaotic system is presented in this paper. (3) the four-dimensional piecewise affine systems The existence of double focus homoclinic orbit. The existence of four with two subsystems of piecewise affine system with dual focus homoclinic orbits, and gives the necessary and sufficient conditions of the switching surface transverse intersection in double homoclinic orbits have the focus, and gives the strict mathematical method of constructing chaotic system. (4) the existence of four piecewise affine system bifocus heteroclinic loop. The existence of four with two subsystems of piecewise affine system with dual focus heteroclinic loop, gives a sufficient and necessary condition of the dual focus heteroclinic and switching surface cross section compared to the two loops exist. And on this basis, we construct a with two different focus homoclinic loop four-dimensional system, gives the chaotic invariant sets are the results of computer simulation. The main contents of the paper are as follows: the first chapter mainly introduces some basic concepts of piecewise smooth systems and piecewise smooth dynamical systems. Research status. The second chapter mainly introduces the symbolic dynamics and topological theory. The third chapter mainly studies the necessary and sufficient conditions for a class of three-dimensional piecewise affine systems exist homoclinic orbit, a mathematical method for constructing chaotic system is presented, and on this basis, the structure of several chaotic systems, the computer simulation results is also given out. The fourth chapter introduces the necessary and sufficient conditions for a class of three-dimensional piecewise affine systems exist heteroclinic loops, and based on the use of topological horseshoe theory to prove the existence of chaotic invariant set, a mathematical method for constructing chaotic system is proposed. And on this basis, the structure of several chaotic systems, the computer simulation results are given related fifth chapter is existence of four-dimensional piecewise affine systems of singular bad. Necessary and sufficient conditions for the first study a class of four-dimensional piecewise affine systems exist double focus homoclinic orbit, a given The mathematical method of chaotic system construction, and based on the structure of a chaotic system, the simulation results are also given. The necessary and sufficient conditions related to study a class of four-dimensional piecewise affine systems bifocus heteroclinic loops exist, created a dual focus heteroclinic loop four-dimensional system, gives the chaotic invariant in the simulation results. The sixth chapter summarizes the whole work, and make the plan for the next step.

【學(xué)位授予單位】：華中科技大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O415.5

【相似文獻(xiàn)】

相關(guān)期刊論文 前10條

1 肖國超 ,張祖勛;仿射變形象片糾正原理及其應(yīng)用[J];測繪學(xué)報;1981年02期

2 谷超豪;娭仿射的安堓UO楲[J];數(shù)學(xué)學(xué)報;1956年03期

3 谷超豪;;^~z.特殊的仿射，

本文編號：1533424