非光滑映射的混沌动力学研究

发布时间：2018-06-21 19:45

本文选题：碰撞振子 + 擦边分岔　；参考：《西南交通大学》2016年博士论文

【摘要】：近年来,非光滑动力系统成为了数学和工程领域研究的一个新热点.一方面,来源于很多实际问题的系统是非光滑的,例如碰撞振动系统,带有干摩擦的粘滑振动系统,含有开关的电路系统以及一些控制系统等;另一方面,许多光滑系统的全局Poincare映射是非光滑的.非光滑系统能够发生一些在光滑系统中不会出现的特有的分岔,例如擦边分岔(Grazing bifurcation),滑移分岔(Sliding bifurcation).同时这些分岔也导致了通向混沌的新路径.第一章综述近年来非光滑动力学的部分结果,最新进展以及尚存在的一些问题.同时,还介绍了本文的研究内容和主要结果.第二章回顾一些将在下文中用到的动力系统和遍历理论的基本概念和结果,包括Birkhoff遍历定理,测度熵,Lyapunov指数,物理测度,Smale马蹄等.第三章研究了一个描述碰撞振子擦边分岔的区间映射的统计性质.首先证明在适当的参数区域内映射是拓扑混合的.接下来对导数进行变形估计(由于该映射具有无界导数的点),在此基础上构造原映射的一个诱导马尔可夫(Markov)映射,并且证明诱导映射的回归时间函数的尾是指数衰减的.然后证明映射存在一个绝对连续不变测度,该测度是唯一的并且混合的.最后应用马尔可夫塔方法证明映射对Holder连续观测满足指数相关性衰减和中心极限定理.第四章证明了在一定的参数区域内,Nordmark映射(单自由度碰撞振子擦边分岔的范式映射)存在马蹄型混沌.首先通过对坐标平面做一个合适的分割,证明Nordmark映射的非游荡集包含在一个矩形区域中,然后从此区域出发构造“横条”和“竖条”,最后验证Conley-Moser条件,证明Nordmark映射在其非游荡集上的限制拓扑共轭于双边符号空间上的移位映射.第五章研究了一类Belykh型映射(一类两维不连续分段线性映射)的符号动力学.首先证明当映射满足双曲性条件时,修剪前猜想(Pruning front conjec-ture) 对此映射在一个由正负向轨道都有界的点构成的不变集上成立,给出了判断系统的允许符号序列的解析条件.在此基础上,我们构造映射的一个拓扑模型,虽然此映射是不连续的,但其在上述不变集上的限制拓扑共轭于双边符号空间的一个商空间上的移位映射.这个商空间由映射的修剪前(Pruning front)和基本修剪区域(Primary pruned region)完全确定.最后我们给出映射存在马蹄型混沌的参数区域的精确边界.第六章继续研究第五章中的Belykh型映射.我们计算这类映射的奇怪吸引子的Hausdorff维数.首先我们证明在一定的参数区域内此映射存在一个捕获域,双曲不动点的不稳定流形包含在捕获域中,所以映射存在奇怪吸引子,然后确定映射存在SRB测度的一个参数区域,通过计算吸引子的容度(盒维数)给出其Hausdorff维数的一个上界,最后应用Young关于Hausdorff维数的公式和Pesin熵公式,给出了吸引子的Hausdorff维数的一个下界.由于上下界相等,所以本文得到了吸引子的Hausdorff维数的精确公式.
[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.
【学位授予单位】：西南交通大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O19

【参考文献】