反演极限与Lauwerier吸引子（Ⅱ）

郭峰; 李登辉

doi:10.3879/j.issn.1000-0887.2014.07.009

反演极限与Lauwerier吸引子（Ⅱ）

doi: 10.3879/j.issn.1000-0887.2014.07.009

郭峰,
李登辉

西南交通大学力学与工程学院, 成都 610031

基金项目: 国家自然科学基金(11172246；11272268)

详细信息

作者简介:
郭峰 (1976—) 男,山东泰安人, 博士生(通讯作者. E-mail: mathguofeng@163.com)

中图分类号: O185.1
计量
- 文章访问数: 815
- HTML全文浏览量: 61
- PDF下载量: 795
- 被引次数: 0
出版历程
- 收稿日期: 2013-12-27
- 修回日期: 2014-04-21
- 刊出日期: 2014-07-15

Inverse Limit and Lauwerier Attractor(Ⅱ)

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, P.R.China

Funds: The National Natural Science Foundation of China(11172246；11272268)

摘要

摘要: 对适当的参数, 二次映射有一条吸引的周期轨道, 并且其吸引集在单位闭区间上是稠密的.根据此性质, 文中定义了Lauwerier映射的一个上半连续分解.在此分解上存在一个可分商空间, 通过投影将二维的Lauwerier映射降为一维的二次映射, 运用二次映射反演极限空间上的移位映射来研究Lauwerier映射的动力学性质.首先对二次映射进行几乎Markov分割, 然后将每个分割区间扩张成相应的小矩形区域, 再对Lauwerier映射进行几乎Markov分割后, 从而证明了当参数小于4时, Lauwerier映射与二次映射反演极限空间上的移位映射是拓扑半共轭的.
- Lauwerier映射 /
- 反演极限空间 /
- 上半连续分解 /
- Markov分割 /
- 拓扑半共轭
Abstract: The quadratic mapping had an attracting periodic orbit of which the attraction set was dense in a unit closed interval for an appropriate parameter. According to that property, an upper semi-continuous decomposition of the Lauwerier mapping was defined, with respect to which there existed a separable quotient space. The 2D Lauwerier mapping was reduced to a 1D quadratic mapping through projection. The dynamic properties of the Lauwerier mapping was studied with the shift map on the inverse limit space of the quadratic mapping. First, the quadratic mapping was nearly Markov partitioned, then each partition interval was expanded to a corresponding small rectangular region, in turn the Lauwerier mapping was nearly Markov partitioned again. It is proved that the Lauwerier mapping is topologically semi-conjugate to the shift map on the inverse limit space of the quadratic mapping when the parameter is under 4.
- Lauwerier mapping /
- inverse limit space /
- upper semi-continuous decomposition /
- Markov partition /
- topologically semi-conjugate

HTML全文

参考文献(18)

[1]	Ruelle D, Takens F. On the nature of turbulence[J]. Commun Math Phys,1971,20(3): 167-192.
[2]	乐源, 谢建华. 一类双面碰撞振子的对称性尖点分岔与混沌[J]. 应用数学和力学, 2007,28(8): 991-998.(YUE Yuan, XIE Jian-hua. Symmetry, cusp bifurcation and chaos of an impact cscillator between two rigid sides[J]. Applied Mathematics and Mechanics,2007,28(8): 991-998.(in Chinese))
[3]	Leine R I, Nijmeijer H. Dynamics and Bifurcations of Non-Smooth Mechanical Systems [M]. Springer, 2004.
[4]	LI Shi-hai. Dynamical properties of the shift maps on the inverse limit space[J]. Ergodic Theory and Dynamical Systems,1992,12(1): 95-108.
[5]	Barge M, Martin J. Chaos, periodicity, and snake-like continua[J]. Trans Amer Math Soc,1985,289(1): 355-365.
[6]	Williams R F. One-dimensional non-wandering sets[J]. Topologically,1967,6(4): 473-487.
[7]	Williams R F. Expanding attractors[J]. Publications Mathématiques de l'Institut des Hautes tudes Scientifiques,1974,43(1): 169-203.
[8]	Williams R F. The structure of attractors[C]//Actes du Congrès International des Mathematics,1970,2: 947-951.
[9]	Williams R F. The structure of Lorenz attractors[J].Publications Mathématiques de l’Institut des Hautes Etudes Scientifiques Turbulence Seminar,1979,50(1): 73-99.
[10]	Guckenheimer J, Williams R F. Structure stability of Lorenz attractors[J]. Publications Mathématiques de l’Institut des Hautes Etudes Scientifiques Turbulence Seminar,1979,50(1): 59-72.
[11]	Lauwerier H A. The structure of a strange attractor[J]. Physica D: Nonlinear Phenomena,1986,21(1): 146-154.
[12]	Liu Z R, Qin W X, Xie H M. The structure and dynamics of Lauwerier attractor[J]. Chinese Science Bulletin,1992,37(14): 1269-1278.
[13]	郭峰, 李登辉, 谢建华. 反演极限与Lauwerier吸引子[J]. 应用数学和力学, 2014,35(2): 212-218.(GUO Feng, LI Deng-hui, XIE Jian-hua. Inverse limits and Lauwerier attractor [J]. Applied Mathematics and Mechanics,2014,35(2): 212-218.(in Chinese))
[14]	Swiatek C. Hyperbolicity is dense in the real quadratic family[R]. Stony Brook Preprint, 1992.
[15]	Holte S, Roe R. Inverse limits associated with the forced Van der Pol equation[J]. Journal of Dynamics and Differential Equations,1994,6(4): 601-612.
[16]	Barge M, Holte S. Nearly one-dimensional Hénon attractors and inverse limits[J]. Nonlinearity,1995,8(1): 29-42.
[17]	Maid R. Hyperbolicity, sinks and measure in one dimensional dynamics[J]. Communications in Mathematical Physics,1985,100(4): 495-524.
[18]	Singer D. Stable orbits and bifurcation of maps of the interval[J]. SIAM Journal on Applied Mathematics,1978,35(2): 260-267.