ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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Isaeva O. B., Kuznetsov S. P. Approximate description of the mandelbrot set. Thermodynamic analogy. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, iss. 1, pp. 55-71. DOI: 10.18500/0869-6632-2006-14-1-55-71

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Russian
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Article
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517.9

Approximate description of the mandelbrot set. Thermodynamic analogy

Autors: 
Isaeva Olga Borisovna, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

Analogy between an approximate version of period-doubling (and period N-tupling) renormalization group analysis in complex domain and the phase transition theory of Yang-Lee (based on consideration of formally complexified thermodynamic values) is discussed. It is shown that the Julia sets of the renormalization transformation correspond to the approximation of Mandelbrot set of the original map. New aspects of analogy between the theory of dynamical systems and the phase transition theory are uncovered.

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Reference: 
  1. Rumer YuB, Ryvkin MSh. Thermodynamics, statistical physics and kinetics. Moscow: Nauka; 1977. 552 p. (In Russian).
  2. Sinai YG. Theory of phase transitions: Strict results. Moscow: Nauka; 1980. 208 p. (In Russian).
  3. Balescu R. Radu Equilibrium and nonequilibrium statistical mechanics. New York: Wiley; 1975. 756 p. (In Russian).
  4. Paitgen H-O, Richter PH. Beauty of fractals. Images of complex dynamic systems. Moscow: Mir; 1993. 176 p. (In Russian).
  5. Kadanoff LP. Scaling laws for Ising models near T(c). Physics. 1966;2(6):263–272. DOI: 10.1103/PHYSICSPHYSIQUEFIZIKA.2.263.
  6. Wilson KG. Renormalisation group and critical phenomena. Phys. Rev. B. 1971;4:3174–3183. DOI: 10.1103/physrevb.4.3174.
  7. Feigenbaum MJ. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 1978;19(1):25–52. DOI: 10.1007/BF01020332.
  8. Feigenbaum MJ. The universal metric properties of non-linear transformations. J. Stat. Phys. 1979;21(6):669–706. DOI: 10.1007/BF01107909.
  9. Schuster H.G. Deterministic Chaos. Weinheim: Physik-Verlag; 1984. 250 p.
  10. Haken G. Synergetics. Berlin, New York: Springer-Verlag, Heidelberg; 1982.
  11. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
  12. Kuznetsov AP, Kuznetsov SP. Critical dynamics of one-dimensional mappings: Part 1. Feigenbaum script. Izvestiya VUZ. Applied Nonlinear Dynamics. 1993;1(1–2):15–33.
  13. Yang CN, Lee TD. Statistical theory of equations of state and phase transitions: 1. Theory of condensation. Phys. Rev. 1952;87(3):404–409. DOI: 10.1103/PHYSREV.87.404.
  14. Lee TD, Yang CN. Statistical theory of equations of state and phase transitions: 2. Lattice gas and Ising model. Phys. Rev. 1952;87(3):410–419. DOI: 10.1103/PHYSREV.87.410.
  15. Wood DW, Turnbull RW. Numerical experiments on Yang-Lee zeros. J. Phys. A: Math. Gen. 1986;19(13):2611–2624. DOI: 10.1088/0305-4470/19/13/026.
  16. Derrida B, De Seze L, Itzykson C. Fractal structure of zeros in hierarchical models. J. Stat. Phys. 1983;33(3):559–569. DOI: 10.1007/BF01018834.
  17. Onsager L. Crystal statistics: 1. Two-dimensional model with an order-disorder transition. Phys. Rev. 1944;65(3-4):117–149. DOI: 10.1103/PHYSREV.65.117.
  18. Wim van Saarloos, Kurtze DA. Location of zeros in the complex temperature plane: Absence of Lee-Yang theorem. J. Phys. A: Math. Gen. 1984;17(6):1301–1311. DOI: 10.1088/0305-4470/17/6/026.
  19. Ananikian NS, Ghulghazaryan RG. Yang-Lee and Fisher zeros of multisite interaction Ising models on the Cayley-type lattices. Phys. Lett. A. 2000;277(4):249–256. DOI: 10.1016/S0375-9601(00)00713-1.
  20. Ananikian NS, Izmailian NSh, Oganessyan KA. An Ising spin-S model on generalized recursive lattice. Physica A. 1998;254(1):207–214. DOI: 10.1016/S0378-4371(98)00013-2.
  21. Ananikian NS, Dallakian SK, Hu B, Izmailian NSh, Oganessyan KA. Chaos in Z(2) gauge model on a generalized Bethe lattice of plaquettes. Phys. Lett. A. 1998;248(5-6):381–385. DOI: 10.1016/S0375-9601(98)00687-2.
  22. Ananikian NS, Dallakian SK. Multifractal approach to three-site antiferromagnetic Ising model. Physica D. 1997;107(1):75–82. DOI: 10.1016/S0167-2789(97)00060-2.
  23. Akheyan AZ, Ananikian NS. Global Rethe lattice consideration of the spin-1 Ising model. J. Phys. A: Math. Gen. 1996;29(4):721–731. DOI: 10.1088/0305-4470/29/4/004.
  24. Landau LD, Lifshits EM. Hydrodynamics. Moscow: Nauka; 1986. 736 p. (In Russian).
  25. Crownover RM. Introduction to Fractals and Chaos. Boston, London: Jones and Bartlett Publishers; 2000. 352 p. (In Russian).
  26. Cvitanovic P, Myrheim J. Complex universality. Commun. Math. Phys. 1989;121(2):225–254.
  27. Isaeva OB, Kuznetsov SP. On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade. Regular and Chaotic Dynamics. 2000;5(4):459–476. DOI: 10.1070/RD2000v005n04ABEH000159.
  28. Gol'berg AI, Sinai YaG, Khanin KM. Universal properties for sequences of bifurcations of period three. Russian Math. Surveys. 1983;38(1):187–188. DOI: 10.1070/RM1983v038n01ABEH003398.
  29. Nauenberg M. Fractal boundary of domain of analyticity of the Feigenbaum function and relation to the Mandelbrot set. J. Stat. Phys. 1987;47(3–4):459–475. DOI: 10.1007/BF01007520.
  30. Buff X. Geometry of the Fiegenbaum map. Conformal Geometry and Dynamics. 1999;3:79–101.
  31. Wells ALJ, Overill RE. The extension of the Feigenbaum-Cvitanovic function to the complex plane. Int. J. of Bif. and Chaos. 1994;4(4):1041–1051.
  32. Widom M, Bensimon D, Kadanoff LP. Strange objects in the complex plane. J. Stat. Phys. 1983;32(3):443–454. DOI: 10.1007/BF01008949.
  33. Jensen MH, Kadanoff LP, Procaccia I. Scaling structure and thermodynamics of strange sets. Phys. Rev. A. 1987;36(3):1409–1420. DOI: 10.1103/physreva.36.1409.
  34. Douady A, Hubbard JH. Iteration des polynomes quadratiques complexes. Paris: CRAS. 1984;294:123–126. (On the dynamics of polynomial-like mappings. Electronic preprint, 1984).
Received: 
25.07.2005
Accepted: 
26.12.2005
Published: 
28.04.2006
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