本文選題：碰撞振子 + 擦邊分岔　； 參考：《西南交通大學(xué)》2016年博士論文

【摘要】：近年來(lái),非光滑動(dòng)力系統(tǒng)成為了數(shù)學(xué)和工程領(lǐng)域研究的一個(gè)新熱點(diǎn).一方面,來(lái)源于很多實(shí)際問(wèn)題的系統(tǒng)是非光滑的,例如碰撞振動(dòng)系統(tǒng),帶有干摩擦的粘滑振動(dòng)系統(tǒng),含有開(kāi)關(guān)的電路系統(tǒng)以及一些控制系統(tǒng)等;另一方面,許多光滑系統(tǒng)的全局Poincare映射是非光滑的.非光滑系統(tǒng)能夠發(fā)生一些在光滑系統(tǒng)中不會(huì)出現(xiàn)的特有的分岔,例如擦邊分岔(Grazing bifurcation),滑移分岔(Sliding bifurcation).同時(shí)這些分岔也導(dǎo)致了通向混沌的新路徑.第一章綜述近年來(lái)非光滑動(dòng)力學(xué)的部分結(jié)果,最新進(jìn)展以及尚存在的一些問(wèn)題.同時(shí),還介紹了本文的研究?jī)?nèi)容和主要結(jié)果.第二章回顧一些將在下文中用到的動(dòng)力系統(tǒng)和遍歷理論的基本概念和結(jié)果,包括Birkhoff遍歷定理,測(cè)度熵,Lyapunov指數(shù),物理測(cè)度,Smale馬蹄等.第三章研究了一個(gè)描述碰撞振子擦邊分岔的區(qū)間映射的統(tǒng)計(jì)性質(zhì).首先證明在適當(dāng)?shù)膮?shù)區(qū)域內(nèi)映射是拓?fù)浠旌系?接下來(lái)對(duì)導(dǎo)數(shù)進(jìn)行變形估計(jì)(由于該映射具有無(wú)界導(dǎo)數(shù)的點(diǎn)),在此基礎(chǔ)上構(gòu)造原映射的一個(gè)誘導(dǎo)馬爾可夫(Markov)映射,并且證明誘導(dǎo)映射的回歸時(shí)間函數(shù)的尾是指數(shù)衰減的.然后證明映射存在一個(gè)絕對(duì)連續(xù)不變測(cè)度,該測(cè)度是唯一的并且混合的.最后應(yīng)用馬爾可夫塔方法證明映射對(duì)Holder連續(xù)觀測(cè)滿足指數(shù)相關(guān)性衰減和中心極限定理.第四章證明了在一定的參數(shù)區(qū)域內(nèi),Nordmark映射(單自由度碰撞振子擦邊分岔的范式映射)存在馬蹄型混沌.首先通過(guò)對(duì)坐標(biāo)平面做一個(gè)合適的分割,證明Nordmark映射的非游蕩集包含在一個(gè)矩形區(qū)域中,然后從此區(qū)域出發(fā)構(gòu)造“橫條”和“豎條”,最后驗(yàn)證Conley-Moser條件,證明Nordmark映射在其非游蕩集上的限制拓?fù)涔曹椨陔p邊符號(hào)空間上的移位映射.第五章研究了一類Belykh型映射(一類兩維不連續(xù)分段線性映射)的符號(hào)動(dòng)力學(xué).首先證明當(dāng)映射滿足雙曲性條件時(shí),修剪前猜想(Pruning front conjec-ture) 對(duì)此映射在一個(gè)由正負(fù)向軌道都有界的點(diǎn)構(gòu)成的不變集上成立,給出了判斷系統(tǒng)的允許符號(hào)序列的解析條件.在此基礎(chǔ)上,我們構(gòu)造映射的一個(gè)拓?fù)淠Ｐ?雖然此映射是不連續(xù)的,但其在上述不變集上的限制拓?fù)涔曹椨陔p邊符號(hào)空間的一個(gè)商空間上的移位映射.這個(gè)商空間由映射的修剪前(Pruning front)和基本修剪區(qū)域(Primary pruned region)完全確定.最后我們給出映射存在馬蹄型混沌的參數(shù)區(qū)域的精確邊界.第六章繼續(xù)研究第五章中的Belykh型映射.我們計(jì)算這類映射的奇怪吸引子的Hausdorff維數(shù).首先我們證明在一定的參數(shù)區(qū)域內(nèi)此映射存在一個(gè)捕獲域,雙曲不動(dòng)點(diǎn)的不穩(wěn)定流形包含在捕獲域中,所以映射存在奇怪吸引子,然后確定映射存在SRB測(cè)度的一個(gè)參數(shù)區(qū)域,通過(guò)計(jì)算吸引子的容度(盒維數(shù))給出其Hausdorff維數(shù)的一個(gè)上界,最后應(yīng)用Young關(guān)于Hausdorff維數(shù)的公式和Pesin熵公式,給出了吸引子的Hausdorff維數(shù)的一個(gè)下界.由于上下界相等,所以本文得到了吸引子的Hausdorff維數(shù)的精確公式.
[Abstract]:In recent years, non smooth dynamic systems have become a new hot spot in the field of mathematics and engineering. On the one hand, systems derived from many practical problems are non smooth, such as collision vibration systems, stick sliding vibration systems with dry friction, circuit systems containing switches and some control systems, on the other hand, many smooth systems. The global Poincare mapping is non smooth. Non smooth systems can have some unique bifurcations that will not appear in smooth systems, such as Grazing bifurcation, and slip bifurcation (Sliding bifurcation). At the same time, these bifurcations also lead to a new path to chaos. Chapter 1 summarizes the part of non smooth dynamics in recent years. In the second chapter, the second chapter reviews the basic concepts and results of the power system and ergodic theory that will be used below, including the Birkhoff ergodic theorem, the measure entropy, the Lyapunov index, the physical measure, the Smale horseshoe, and so on. The third chapter studies A statistical property that describes the interval mapping of the edge splitting bifurcation of the collision oscillator is given. First, it is proved that the mapping is a topological mixture in the proper parameter region. Then the derivative is estimated (because the map has the point of unbounded derivative), and on this basis, a induced Markov (Markov) mapping of the original mapping is constructed and the lure is proved. The tail of the regression time function of the guided mapping is exponential decay. Then it is proved that the mapping has an absolute continuous invariant measure, which is unique and mixed. Finally, the Markov tower method is used to prove that the mappings satisfy the exponential correlation attenuation and the central limit theorem for Holder continuous observation. The fourth chapter proves that the parameters are in certain parameters. In the region, there is a horseshoe type chaos in the Nordmark mapping (the paradigm mapping of the single degree of freedom collisions). First, it is proved that the non wandering set of the Nordmark mapping is included in a rectangular region by a proper segmentation of the coordinate plane, and then the "bar" and the "vertical bar" are constructed from the region, and the Conley-Moser is finally verified. The fifth chapter studies the symbolic dynamics of a class of Belykh type mappings (a class of two dimensional discontinuous piecewise linear mappings). The first is to prove that when the mappings satisfy the hyperbolic strip, the pre pruning conjecture (Pruning front conjec-ture) is proved. On this basis, we construct a topological model of the mapping, although the mapping is discontinuous, but the restricted topology on the above invariant sets is conjugated to the bilateral symbolic space. A shift mapping on a quotient space. The quotient space is completely determined by the mapped pruning (Pruning front) and the basic pruning region (Primary pruned region). Finally, we give the exact boundary of the parameter region of the mapped horseshoe type chaos. The sixth chapter continues to study the Belykh mapping in the fifth chapter. We calculate this kind of mapping. The Hausdorff dimension of the odd attractor. First, we prove that there is a capture domain in a certain parameter region, and the unstable manifold of the hyperbolic fixed point is included in the capture domain, so the mapping has strange attractors, and then a parameter region of the mapping has a SRB measure, and the volume of the attractor (box dimension) is calculated. In this paper, a upper bound of the Hausdorff dimension is given. Finally, a lower bound for the Hausdorff dimension of the attractor is given by using the formula of Young's Hausdorff dimension and the Pesin entropy formula. The exact formula of the Hausdorff dimension of the attractor is obtained because the upper and lower bounds are equal.
【學(xué)位授予單位】：西南交通大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O19

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