Labyrinthine pathways towards supercycle attractors in unimodal maps

• Authors: Luis Moyano , D. Silva , A. Robledo
• Languages of publication: EN

Abstracts

EN

As an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.

Keywords

EN

supercycles; fractal boundaries; transient chaos; RG fixed-point map

Discipline

• 02.50.-r: Probability theory, stochastic processes, and statistics(see also section 05 Statistical physics, thermodynamics, and nonlinear dynamical systems)
• 02.70.Rr: General statistical methods
• 89.75.Da: Systems obeying scaling laws

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2009
• Volume: 7
• Issue: 3
• Pages: 591-600

Dates

published

1 - 9 - 2009

online

25 - 6 - 2009

Contributors

author

Luis Moyano

• Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos, Universidad Carlos III de Madrid, 28911, Leganés, Madrid, Spain

author

D. Silva

• Instituto de Física, Universidad Nacional Autónoma de México, Apartado postal 20-364, México, 01000 D.F., México

author

A. Robledo

• Instituto de Física, Universidad Nacional Autónoma de México, Apartado postal 20-364, México, 01000 D.F., México

References

• [1] H. G. Schuster, Deterministic Chaos. An Introduction, 2nd Revised Edition (VCH Publishers, Weinheim, 1988)
• [2] A. Robledo, Europhysics News 36, 214 (2005) http://dx.doi.org/10.1051/epn:2005611[Crossref]
• [3] A. Robledo, arXiv:0710.1047 [WoS]
• [4] P. Grassberger, R. Badii, A. Politi, J. Stat. Phys. 51, 135 (1988) http://dx.doi.org/10.1007/BF01015324[Crossref]
• [5] H. Mori, H. Hata, T. Horita, T. Kobayashi, Prog. Theor. Phys. Supp. 99, 1 (1989) http://dx.doi.org/10.1143/PTPS.99.1[Crossref]
• [6] D. J. Evans, G. P. Morris, Statistical Mechanics of Nonequilibrium Liquids, Theoretical Chemistry Monograph Series (Academic Press, London, 1990)
• [7] C. Beck, F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, UK, 1993) http://dx.doi.org/10.1017/CBO9780511524585[Crossref]
• [8] R. C. Hilborn, Chaos and Nonlinear Dynamics, 2nd Revised Edition (Oxford University Press, New York, 2000) http://dx.doi.org/10.1093/acprof:oso/9780198507239.001.0001[Crossref]
• [9] A. Robledo, Physica A 314, 437 (2002) http://dx.doi.org/10.1016/S0378-4371(02)01177-9[Crossref]
• [10] A. Robledo, Physica D 193, 153 (2004) http://dx.doi.org/10.1016/j.physd.2004.01.016[Crossref]
• [11] F. Baldovin, A. Robledo, Europhys. Lett. 60, 518 (2002) http://dx.doi.org/10.1209/epl/i2002-00249-7[Crossref]
• [12] B. Hu, J. Rudnick, Phys. Rev. Lett. 48, 1645 (1982) http://dx.doi.org/10.1103/PhysRevLett.48.1645[Crossref]
• [13] C. Grebogi, S. W. McDonald, E. Ott, J. A. Yorke, Phys. Lett. 99A, 415 (1983)
• [14] S. W. McDonald, C. Grebogi, E. Ott, J. A. Yorke, Physica D 17, 125 (1985) http://dx.doi.org/10.1016/0167-2789(85)90001-6[Crossref]
• [15] E. Mayoral, A. Robledo, Phys. Rev. E 72, 026209 (2005) http://dx.doi.org/10.1103/PhysRevE.72.026209[Crossref]
• [16] H. Hernández-Saldaña, A. Robledo, Physica A 370, 286 (2006) http://dx.doi.org/10.1016/j.physa.2006.03.018[Crossref]
• [17] A. Robledo, Physica A 370, 449 (2006) http://dx.doi.org/10.1016/j.physa.2006.06.003[Crossref]
• [18] A. Robledo, L. G. Moyano, arXiv:0706.4422 [WoS]
• [19] D. Sornette, Phys. Rep. 297, 239 (1998) http://dx.doi.org/10.1016/S0370-1573(97)00076-8[Crossref]
• [20] A. Robledo, L. G. Moyano, Phys. Rev. E 77, 032613 (2008) http://dx.doi.org/10.1103/PhysRevE.77.036213[Crossref]

Identifiers

DOI

10.2478/s11534-009-0065-1