Symmetries of star products and metric universalities in 1D quadri-modal maps

Chaos, Solitons and Fractals

Volumn 20, Issue 3, 2004, Pages 547-560

  Zhou, Zhong ^a   Gao, Wen ^a   Liu, Hong Zhang ^a   Peng, Shou Li ^a,b  

^a YUNNAN UNIVERSITY   (China)

^b CCAST WORLD LABORATORY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

ENTROPY; NUMERICAL METHODS; TOPOLOGY;

QUADRI-MODAL MAPS;

BIFURCATION (MATHEMATICS);

EID: 0242365781     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0960-0779(03)00413-2     Document Type: Article

References (33)

1. Metropolis N., Stein M.L., Stein P.R. On finite limit sets for transformations on the unit interval. J. Comb. Theory A. 15:1973;25-44.
2. Feigenbaum M.J. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19:1978;25-52.
3. Feigenbaum M.J. The universal metric properties of nonlinear transformations. J. Stat. Phys. 21:1979;669-706.
4. Derrida B., Gervois A., Pomeau Y. Iteration of endomorphisms on the real axis and representation of numbers. Ann. Inst. Henri Poincaré A. 29:1978;305-356.
5. Milnor J., Thurston W. Alexander J.C. On iterated maps of the interval. Lecture Notes in Mathematics. vol. 1342:1988;465-563 Springer-Verlag, Berlin.
6. Chang S.J., Wortis M., Wright J.A. Iterative properties of a one-dimensional quartic map: critical lines and tricritical behavior. Phys. Rev. A. 24:1981;2669-2683.
7. Brucks K., Galeeva R., Mumbrú P., Rockmore D., Tresser C. On the *-product in kneading theory. Fundam. Math. 152:1997;189-209.
8. Mackay R.S., Tresser C. Some fleshes on the skeleton: the bifurcation structure of bimodal maps. Physica D. 27:1987;412-422.
9. Hao B.L. Elementary symbolic dynamics and chaos in dissipative systems. 1989;World Scientific, Singapore. 105-11.
10. Hao B.L. Symbolic dynamics and characterization of complexity. Physica D. 51:1991;161-176.
11. Xie F.G., Hao B.L. Symbolic dynamics of the sine-square map. Chaos, Solitons & Fractals. 3:1993;47-60.
12. Ringland J. A genealogy for the periodic orbits of a class of 1D maps. Physica D. 79:1994;289-298.
13. Kurths J., Schwarz U., Witt A. Nonlinear data analysis in solar radio astronomy. Lecture Notes in Physics. 444:1995;159-171.
14. Saparin P., Kurths W., Kurths J., Felsenberg D. Quantification of cancellous bone structure using symbolic dynamics and measures of complexity. Phys. Rev. E. 58:1998;6449-6459.
15. Bollt E., Stanford T., Lai Y.C., Zyczkowski K. Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series. Phys. Rev. Lett. 85:2000;3524-3527.
16. Bollt E., Jones M.A. Developing symbolic dynamics to measure the complexity of repeated game strategies by topological entropy. Nonlinear Dynam., Psychol., Life Sci. 4:2000;153-168.
17. Kaplan H. New method for calculating stable and unstable periodic orbits of one-dimensional maps. Phys. Lett. A. 97:1983;365.
18. Zeng W.Z., Hao B.L., Wang G.R., Chen S.G. Scaling property of period-. n -tupling sequences in one-dimensional mappings Commun. Theor. Phys. 3:1984;283.
19. Zhou Z., Cao K.F. An effective numerical method of the word-lifting technique in one-dimensional multimodal maps. Phys. Lett. A. 310:2003;52-57.
20. Peng S.L., Cao K.F., Chen Z.X. Devil's staircase of topological entropy and global metric regularity. Phys. Lett. A. 193:1994;148-154.
21. Cao K.F., Chen Z.X., Peng S.L. Global metric regularity of the devil's staircase of topological entropy. Phys. Rev. E. 51:1995;1989-1995.
22. Liu H.Z., Zhou Z., Peng S.L. Star transformations and their genealogical varieties in symbolic dynamics of four letters. J. Phys. A: Math. Gen. 31:1998;8431-8450.
23. Peng S.L., Zhang X.S., Cao K.F. Dual star products and metric universality in symbolic dynamics of three letters. Phys. Lett. A. 246:1998;87-96.
24. Peng S.L., Du L.M. Dual star products and symbolic dynamics of Lorenz maps with the same entropy. Phys. Lett. A. 261:1999;63-73.
25. Cao K.F., Peng S.L. Complexity of routes to chaos and global regularity of fractal dimensions in bimodal maps. Phys. Rev. E. 60:1999;2745-2760.
26. Peng S.L., Zhang X.S. The generalized Milnor-Thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters. Commun. Math. Phys. 213:2000;381-411.
27. Zhou Z., Peng S.L. Cyclic star products and universalities in symbolic dynamics of trimodal maps. Physica D. 140:2000;213-226.
28. Cao K.F., Zhou Z., Gao W., Peng S.L. General form of superuniversality for fractal dimensions in one-dimensional maps. Int. J. Mod. Phys. B. 15:2001;4183-4197.
29. Zhou Z., Peng S.L. Star products of quartic maps with two equally high peaks. Chin. Phys. Lett. 17:2000;252-254.
30. Arrays M., Kaufman I.K., Luchinsky D.G., McClintock P.V.E., Soskin S.M. The Kramers problem for a multi-well potential. Phys. Rev. Lett. 84:2000;2556-2559.
31. Teodoroy F.D., Cerboneschi E., Hennequinz D., Arimondo E. Multi-stability for an extended-cavity diode laser with intracavity atomic absorber. Quant. Semiclass. Opt. 9:1997;867-878.
32. Liu HZ, Peng SL, Zhou Z. A proof of cyclic star products in 1D quintic maps. Preprint 2003.
33. GMP library, Available from: http://swox.com/gmp.