Introduction of Topological Horseshoe Theory in Chaotic Research

Abstract:

Over the past 10 years, nonlinear dynamics and chaotic theory attracted scholars and people got a deeper understanding of chaos. There are many methods for chaos research, and the method of using topological horseshoe is an important branch of those methods. So far, this is one of the core methods with mathematical rigor for chaos research. Based on simple thinking of geometric space, topological horseshoe build a bridge for numerical and theoretical studies of complex behavior of nonlinear systems so that people can carry out a series of studies for chaotic behavior. This paper introduces the basic content of topological horseshoe theory and the application to a simple power system.

• Periodical: Advanced Materials Research (Volume 811)
• Pages: 716-719
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.811.716
• Online since: September 2013
• Authors: You Jie Ma, Shuang Song, Xue Song Zhou
• Keywords: Chaotic Theory, Geometric Theory, Nonlinear Dynamics, Poincare Sections, Topological Entropy, Topological Horseshoe

References

[1] Jia Hong-jie, Yu Yin-xin, Wang Chen-Shan. Chaotic Phenomena in Power Systems and its Studies. Proceedings of the CSEE. 2001, 21(7).

[2] Li Qing-du, Yang Xiao-song. A Simple Method for Finding Topological horseshoes. International Journal of Bifurcation and Chaos. 2010, 20(2).

DOI: 10.1142/s0218127410025545

[3] E N Lorenz. Deterministic nonperiod, flow. Journal of the Atmospheric Sciences. 1963, 20.

[4] Chen G R; Ueta T. International Journal of Bifurcation and Chaos. (1999).

[5] Lu Jin-hu, Chen Guan-rong, Cheng Dai-zhan, Sergej Celikovsky. Bridge the gap between the Lorenz system and the Chen system. International journal of bifurcation and chaos in applied sciences and engineering. 2002, 12(12).

DOI: 10.1142/s021812740200631x

[6] Liu CX, Liu T, Liu L, Liu K. A new chaotic attractor. Chaos, Solitons and Fractals. 2004, 22(5).

DOI: 10.1016/j.chaos.2004.02.060

[7] Qi GY, Chen GR, Du SZ, Chen ZQ, Yuan ZZ. Analysis of a new chaotic system. Physica. A, Statistical & Theoretical Physics. 2005, 352[2-4].

[8] Udaltsov V S, Goedgebuer J P, Larger L, Cuenot J B, Rhodes W T. Optics and Spectroscopy. (2003).

[9] Jin-Yuan Hsieh, Chi-Chuan Hwang, An-Pei Wang, Woei-Jong LiM. Controlling hyperchaos of the Rossler systems. International Journal of Control. 1999, 72(10).

[10] Song YZ. Chinese Physics. (2007).

[11] Liang Song, Zhu Hong-Liang, Pan Jiao-Qing, Wang Wei. Dependence of bimodal size distribution on temperature and optical properties of InAs quantum dots grown on vicinal GaAs (100) substrates by using MOCVD. Chinese physics. 2006, 15(5).

DOI: 10.1088/1009-1963/15/5/042

[12] Wiggins S. Introduction to Applicd Nonlinear Dynamical Systems and Chaos. New York: springer-verlag. (1990).

[13] Morse M, G A Hedlund. Symbolic dynamics. American Journal of Mathematics. 1938, 60(4).

[14] Kennedy J, Yorke J A. Topological horseshoes. Transac-tions of the American Mathematical Society. 2001, 353(6).

[15] Yang X S, Tang Y. Horseshoes in piecewise continuous maps. Chaos, Solitons&Fractals. 2004, 19(4).

DOI: 10.1016/s0960-0779(03)00202-9

[16] Wu  Wenjun, Meng  Fanshun, Li Jing. Co-sensitization with near-IR absorbing cyanine dye to improve photoelectric conversion of dye-sensitized solar cells. Synthetic Metals. 2009, 159(11).

DOI: 10.1016/j.synthmet.2009.01.023

[17] Li Qing-du, Yang Xiao-song. Progresses on Chaotic Dynamics Study with Topological horseshoes. Journal of Dynamics and Control. 2012, 10(4).