Inverse Limit and Lauwerier Attractor(Ⅱ)

GUO Feng; LI Deng-hui

doi:10.3879/j.issn.1000-0887.2014.07.009

Volume 35 Issue 7

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2014 > 35(7): 798-804

GUO Feng, LI Deng-hui. Inverse Limit and Lauwerier Attractor(Ⅱ)[J]. Applied Mathematics and Mechanics, 2014, 35(7): 798-804. doi: 10.3879/j.issn.1000-0887.2014.07.009

Citation:

GUO Feng, LI Deng-hui. Inverse Limit and Lauwerier Attractor(Ⅱ)[J]. Applied Mathematics and Mechanics, 2014, 35(7): 798-804. doi: 10.3879/j.issn.1000-0887.2014.07.009

Citation:

GUO Feng, LI Deng-hui. Inverse Limit and Lauwerier Attractor(Ⅱ)[J]. Applied Mathematics and Mechanics, 2014, 35(7): 798-804. doi: 10.3879/j.issn.1000-0887.2014.07.009

PDF( 427 KB)

Inverse Limit and Lauwerier Attractor(Ⅱ)

doi: 10.3879/j.issn.1000-0887.2014.07.009

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

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

Received Date: 2013-12-27
Rev Recd Date: 2014-04-21
Publish Date: 2014-07-15

Abstract

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

FullText(HTML)

References(18)

References

[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.