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2009 | 7 | 3 | 591-600
Tytuł artykułu

Labyrinthine pathways towards supercycle attractors in unimodal maps

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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.
Wydawca

Czasopismo
Rocznik
Tom
7
Numer
3
Strony
591-600
Opis fizyczny
Daty
wydano
2009-09-01
online
2009-06-25
Twórcy
autor
  • Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos, Universidad Carlos III de Madrid, 28911, Leganés, Madrid, Spain, lmoyano@math.uc3m.es
autor
  • Instituto de Física, Universidad Nacional Autónoma de México, Apartado postal 20-364, México, 01000 D.F., México
autor
  • Instituto de Física, Universidad Nacional Autónoma de México, Apartado postal 20-364, México, 01000 D.F., México, robledo@fisica.unam.mx
Bibliografia
  • [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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.-psjd-doi-10_2478_s11534-009-0065-1
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