Relation between Topological Entropies and Fractal Dimensions of Chaotic Attractors of the Hénon Map

Abstract:

In two-dimensional chaotic dynamics, relationship between fractal dimensions and topological entropies is an important issue to understand the chaotic attractors of Hénon map. we proposed a efficient approach for the estimation of topological entropies through the study on the integral relationship between fractal dimensions and topological entropies. Our result found that there is an approximate linear relation between their topological entropies and fractal dimensions.

• Periodical: Advanced Materials Research (Volume 569)
• Pages: 447-450
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.569.447
• Online since: September 2012
• Authors: Xiao Zhou Chen, Liang Lin Xiong, Long Li
• Keywords: Chaotic Attractor, Fractal Dimension, Hénon Map, Topological Entropy

References

[1] N. Metropolis, PR. Stein, On finite limite sets for transfomations on the unit interval. J Combin Theory A 1973; 15: 24-44.

[2] MJ. Feigenbaum, Quantitative universality for a class of nonlinear transfomations. J Stat Phys 1978; 19: 25-52.

[3] MJ. Feigenbaum, The universal metric properties of nonlinear transfomations. J Stat Phys 1979; 21: 669-706.

[4] B-L Hao, Elementary symbolic dynamics and chaos in dissipative systems. Singapore: Word Scientific; (1989).

[5] B-L Hao, W-M Zheng, Applied symbolic dynamics and chaos. In: Directions in chaos, vol. 7. Singapore: Word Scientific; (1998).

[6] K-F Cao, Z Zhou, W Gao, et al., General form of superuniversality for fractal dimensions in one-dimensional maps [J]. Int J Mod Phys B, 2001, 15(32): 4183-4197.

DOI: 10.1142/s0217979201007956

[7] W-B Zhai, X-Z Chen, K-F Cao, Global regularity of fractal dimensions in quadrumodal maps[J]. Journal of Yunnan University(Natural Sciences Edition), 2004, 26(6): 521-524.

[8] X-S Zhang, X-D Liu, K.H. Kwek, S-L Peng, Disorder versus order: global multifractal relationship between topological entropies and universal convergence rates [J]. Phys Lett A, 1996, 211(3): 148-54.

DOI: 10.1016/0375-9601(95)00923-x

[9] W-B Zhai, X-Z Chen, K-F Cao, Global multifractal relation between topological entropies and fractal dimensions[J]. Chaos, Solitons & Fractals, 2004, 23(2): 511-518.

DOI: 10.1016/j.chaos.2004.05.036

[10] S-L Peng, K-F Cao, Z-X Chen. Devil's stair-case of topological entropy and global metric regularity [J]. Phys Lett A, 1994, 193(5-6): 437-443; 1995, 196(5-6): 378.

DOI: 10.1016/0375-9601(94)00882-p

[11] K-F Cao, X-S Zhang, Z Zhou, et al., Devil's carpet of topological entropy and complexity of global dynamical behacior [J]. Chaos, Solitond & Fractals, 2003, 16(5): 709-726.

DOI: 10.1016/s0960-0779(02)00405-8

[12] L-S Young, Dimension, entropy and Lyapunov exponents [J]. Ergod Theory Dynam Syst, 1982, 2: 109-124.

DOI: 10.1017/s0143385700009615

[13] J.L. KAPLAN, J.A. YORKE, Lecture Notes in Mathematics 730, ed. H D Peitgen, H O Walter, Berlin: Springer, 1979, p.204.

[14] P. SCHMELCHER, F.K. DIAKONOS, Detecting unstable periodic orbits of chaotic dynamical systems [J]. Phys Rev Lett, 1997, 78(25): 4733-4736.

DOI: 10.1103/physrevlett.78.4733