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

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Kuznetsov A. P., Kuznetsov S. P., lvankov N. Y., Osin A. A. Scaling at the transition to chaos via destruction of quasiperiodic motion at the golden mean frequency ratio. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 4, pp. 3-24. DOI: 10.18500/0869-6632-2000-8-4-3-24

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Scaling at the transition to chaos via destruction of quasiperiodic motion at the golden mean frequency ratio

Kuznetsov Aleksandr Petrovich, 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
lvankov Nikolay Yurevich, Saratov State University
Osin Alexei Andreevich, Saratov State University

The paper reproduces basic results concerning transition to chaos via destruction of quasiperiodic motion in one-dimensional circle map. For the golden mean rotation number we present the renormalization group (RG) analysis, consider structure of the critical attractor, give illustrations of scaling in the parameter space of the model map. 
We outline the fact that the two-parameter scaling can be observed only in a specially chosen local coordinates connected with the parameters of the original map via a nonlinear variable change; its form must account the concrete relation between two relevant eigenvalues of the linearized RG equation.

Key words: 
The work was supported by the RFBR, grant № 00-02-17509 and Ministry of Education of the Russian Federation, grant № 97-0-8.3-88.
  1. Landau LD. On the problem of turbulence. C. R. Acad. Sci. URSS. 1944;44: 311-314.
  2. Ruelle D, Takens F. On the nature of turbulence. Commun. Math. Phys. 1971;20:167-192.
  3. Grebogi C, Ott E, Yorke JA. Attractors on an N—torus. Quasiperiodicity versus chaos. Physica D. 1985;15(3):354-373. DOI: 10.1016/S0167-2789(85)80004-X.
  4. Grebogi C, Ott E, Pelikan S, Yorke JA. Strange attractors that are not chaotic. Physica D. 1984;13(1-2):261-268.
  5. Kim Sh, Ostlund S. Simultaneous rational approximations in the study оf dynamical systems. Phys. Rev. A. 1986;34(4):3426-3434. DOI: 10.1103/physreva.34.3426.
  6. Rockmore D, Siegel R, Tongring N, Tresser C. An approach to renormalization on the n—torus. CHAOS. 1991;1(1):25-30. DOI: 10.1063/1.165814.
  7. Baladi V, Rockmore D, Tongring N, Tresser С. Renormalization on the n— dimensional torus. Nonlinearity. 1992;5(5):1111-1137. DOI: 10.1088/0951-7715/5/5/005.
  8. Baesens C, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos. Physica D. 1991;49(3):387.-475. DOI: 10.1016/0167-2789(91)90155-3.
  9. Arnold VI. Cardiac arrythmias and circle mappings. CHAOS. 1991;1(1):20-24.              DOI: 10.1063/1.165812.
  10. Arnold VI. Additional chapters of the theory of ordinary differential equations. М.: Nauka; 1978. 304 p.
  11. Schuster G. Deterministic Chaos: An Introduction. Weinheim: Wiley; 1998. 270 p.
  12. Shenker SJ. Scaling behavior in a map of a circle onto itself: Empirical results. Physica D. 1982;5(2-3):405-411. DOI: 10.1016/0167-2789(82)90033-1.
  13. Feigenbaum MJ, Kadanoff LP, Shenker SJ. Quasiperiodicity in dissipative systems: A renormalization group analysis. Physica D. 1982;5(2-3):370-386. DOI: 10.1016/0167-2789(82)90030-6.
  14. Rand Р, Ostlund S, Sethna J, Siggia ED. Universal transition from quasiperiodicity to chaos in dissipative systems. Phys. Rev. Lett. 1982;49(2):132-135. DOI: 10.1103/PhysRevLett.49.132.
  15. Ostlund S, Rand D, Sethna J, Siggia ED. Universal properties оf the transition from quasiperiodicity to chaos in dissipative systems. Physica D. 1983;8(3):303-342. DOI:10.1016/0167-2789(83)90229-4.
  16. Daido H. On the scaling behavior in а map оf а circle onto itself. Progr. Theor. Phys. 1982;68(6):1935-1940. DOI: 10.1143/PTP.68.1935.
  17. Jensen MH, Bak Р, Bohr Т. Complete devil’s staircase, fractal dimension, and universality оf mode—locking structure in the circle map. Phys. Rev. Lett. 1983;50(21):1637-1639. DOI: 10.1103/PhysRevLett.50.1637.
  18. Bohr T, Bak P, Jensen MH. Transition to chaos by interaction of resonances in dissipative systems. II. Josephson junctions, charge—density waves, and standard maps. Phys. Rev. A. 1984;30(4):1970-1981. DOI: 10.1103/PhysRevA.30.1970.
  19. Jensen MH, Bak Р, Bohr Т. Transition to chaos by interaction оf resonances in dissipative systems. I. Circle maps. Phys. Rev. A. 1984;30(4):1960-1969. DOI: 10.1103/PhysRevA.30.1960.
  20. Bohr T, Gunaratne G. Scaling for supercritical circle maps: numerical investigation of the onset of bistability and period doubling. Phys. Lett. A. 1985;113(2):55-60. DOI: 10.1016/0375-9601(85)90651-6.
  21. Sinaj YaG, Khanin KM, Shur LN. A new approach to the construction of fixed points  of renormalization group transformations in dynamical systems. Radiophysics and Quantum Electronics. 1986;29(9):1061-1066. (in Russian).
  22. Rand D. Fractal bifurcation sets, renormalization strange sets, and their universal invariants. Proc.Roy.Soc. Lond. A. 1987;413(1844):45-61. DOI: 10.1098/rspa.1987.0099.
  23. Wang X, Mainieri R, Lowenstein JH. Circle-map scaling in а two— dimensional setting. Phys.Rev. A. 1989;40(9):5382-5389. DOI: 10.1103/physreva.40.5382.
  24. Kim S, Ostlund S. Universal scaling in circle maps. Physica D. 1989;39(2-3):365-392. DOI: 10.1016/0167-2789(89)90017-1.
  25. Hu B, Valinia А, Piro О. Universality and asymptotic limits of the scaling exponents in circle maps. Phys.Lett. A. 1990;144(1):7-10. DOI: 10.1016/0375-9601(90)90038-P.
  26. Cvitanovic Р, Gunaratne GH, Vinson MJ. On the mode—locking universality for critical circle map. Nonlinearity. 1990;3(3):873-885.
  27. Fourcade B, Tremblay A—MS. Universal multifractal properties оf circle maps from the point оf view оf critical phenomena. I. Phenomenology. J. Stat. Phys. 1990;61(3-4):607-637. DOI: 10.1007/BF01027294.
  28. Fourcade B, Tremblay A—MS. Universal multifractal properties оf circle maps from the point оf view оf critical phenomena. II. Analytical results. J. Stat. Phys. 1990;61(3—4):639-665. DOI: 10.1007/BF01027295.
  29. Christos GA, Cherghetta T. Trajectory scaling functions for the circle map and the quasi-periodic route to chaos. Phys.Rev. A. 1991;44(2):898-907. DOI: 10.1103/PhysRevA.44.898.
  30. Khanin KM. Universal estimates for critical circle mappings. CHAOS. 1991;1(2):181-186. DOI: 10.1063/1.165826.
  31. Pikovsky AS, Zaks MA. On the global scaling properties of mode—lockings in а critical circle map. Phys Lett. A. 1991;155(6-7):373-376. DOI: 10.1016/0375-9601(91)91042-C.
  32. Pikovsky AS, Zaks MA. Farey level separation in mode-locking structure of circle mappings. Physica D. 1992;59(1-3):255-269. DOI: 10.1016/0167-2789(92)90218-C.
  33. Ketoja JA. Renormalization in a circle map with two inflection points. Physiса D. 1992;55(1-2):45-68. DOI: 10.1016/0167-2789(92)90187-R.
  34. MacKay RS. Renormalization of bicritical circle maps. Phys. Lett. A. 1994;187:391-396.
  35. Campbell DK, Galeera R, Tresser C, Uherka DR. Piecewise linear models for the quasiperiodic transition to chaos. CHAOS. 1996;6(2):121-154. DOI: 10.1063/1.166159.
  36. De Spinadel VW. On characterization of the onset to chaos. Chaos, Solitons & Fractals. 1997;8(10):1631-1643. DOI: 10.1016/S0960-0779(97)00001-5.
  37. Ketoja JA, Satija II. Harper equation, the dissipative standard map and strange nonchaotic attractors: Relationship between an eigenvalue problem and iterated maps. Physica D. 1997;109(1-2):70-80. DOI: 10.1016/S0167-2789(97)00160-7.
  38. Dixon TW, Kenny BG. Transition to criticality in circle maps at the golden mean. J. Math. Phys. 1998;39(11):5952-5963. DOI: 10.1063/1.532607.
  39. Dixon TW, Gherghetta T, Kenny BG. Universality in the quasiperiodic route to chaos. CHAOS. 1996;6(1):32-42. DOI: 10.1063/1.166155.
  40. Feigenbaum MJ. Quantitative universality for а class оf nonlinear transformations. J. Stat. Phys. 1978;19:25-52. DOI: 10.1007/BF01020332.
  41. Feigenbaum MJ. Universal behavior in nonlinear systems. Physica D. 1983;7(1-3):16-39. DOI: 10.1016/0167-2789(83)90112-4.
  42. Wul EV, Sinaj YaG, Khanin KM. Feigenbaum’s versatility and thermodynamic formalism. Russian Matematical Surveys. 1984;39(3):3-37. (in Russian).
  43. Kuznetsov AP, Kuznetsov SP. Critical dynamics for one-dimensional maps. Part 1: Feigenbaum’s scenario. Izvestiya VUZ. Applied Nonlinear Dynamics. 1993;1(1):15-33. (in Russian).
  44. Peitgen Н-О, Jiirgens H, Saupe D. Chaos and Fractals. New Frontiers of Science. Berlin, N.Y.: Springer; 1992. 864 p.
  45. Kuznetsov АР, Kuznetsov SP, Sataev IR, Chua LО. Two—parameter study of transition to chaos in Chua’s circuit: renormalization group, universality and scaling. Int. J. of Bif. and Chaos. 1993;3(4):943-962. DOI: 10.1142/S0218127493000799.
  46. Kuznetsov АР, Kuznetsov SP, Sataev IR, Chua LО. Multi—parameter criticality in Chua’s circuit at period—doubling transition to chaos. Int. J. of Bif. and Chaos. 1996;6(1):119-148. DOI: 10.1142/S0218127496001880.
  47. Anishchenko VS, Vadivasova TE, Actakhov VV. Nonlinear dynamics of chaotic and stochastic systems. Saratov: Saratov University Publishing; 1999. 368 p. (in Russian).
  48. Kuznetsov АР, Kuznetsov SP, Sataev IR. A variety of period—doubling universality classes in multi—parameter analysis оf transition to chaos. Physica D. 1997;109:91-112. DOI: 10.1016/S0167-2789(97)00162-0.
  49. Kuznetsov АР, Kuznetsov SP, Sataev IR. Codimension and typicality in the context of the problem of describing the transition to chaos through period doubling in dissipative dynamical systems. Regul. Chaotic Dyn. 1997;2(3-4):90-105. DOI: 10.1070/RD1997v002n04ABEH000050. (in Russian).
  50. Halsey TC, Jensen MH, Kadanoff LP, Procaccia I, Shraiman BI. Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A. 1986;33:1141-1151. DOI: 10.1103/PhysRevA.33.1141.
  51. Beck C, Schlogl Е. Thermodynamics of chaotic systems. Cambridge: Cambridge Univ. Press; 1993, 281 p.
  52. Stavans J, Helsot F, Libchaber A. Fixed winding number and the quasiperiodic route to chaos in а convective fluid. Phys.Rev.Lett. 1985;55:596-599. DOI: 10.1103/PhysRevLett.55.596.
  53. Levi BG. New Global Fractal Formalism Describes Paths to Turbulence. Physics Today. 1986;39(4):17-18. DOI: 10.1063/1.2814964.
  54. Anishchenko VS. Complex oscillations in simple systems. М.: Nauka; 1990. 312 p. (in Russian).