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

For citation:

Kuznetsov S. P., Majlybaev A. A., Sataev I. R. Bifurcation of universal regimes at the border of chaos. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, iss. 3, pp. 27-47. DOI: 10.18500/0869-6632-2005-13-3-27-47

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2005no3p027.pdf
(downloads: 228)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2005-13-3-27-47

Bifurcation of universal regimes at the border of chaos

Autors: 
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Majlybaev Aleksej Abaevich, State Educational Scientific Institution "Research Institute of Mechanics of Moscow State University"
Sataev Igor Rustamovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

It is shown that a fixed point of the renormalization group transformation for a system of two subsystems with unidirectional coupling, one represented by a unimodal map with extremum of degree κ and another by a map accumulating a sum of terms expressed as a function of a state of the first subsystem, undergoes a period-doubling bifurcation in a course of increase of the parameter κ. At κ = 2 the respective solution (period-2 cycle of the renormalization group equation) corresponds to a situation at the chaos threshold designated as the C-type critical behavior (Kuznetsov and Sataev, Phys. Lett., 1992, 236). On a basis of combination of analytic considerations and numerical computations, we construct and analyze an asymptotical expansion of the solution over powers of deflection of the parameter κ from the critical value κc = 1, 984396. The approach is analogous to that known in the phase transition theory as ε-expansion, which relates to presence of a bifurcation from a «trivial» fixed point of renormalization group transformation to a new fixed point, responsible for critical behavior with nontrivial critical indices. 

Key words: 
-
Reference: 

        

  1. Feigenbaum MJ. Universal behavior in nonlinear systems. Physica D. 1983;7(1–3):16–39. DOI: 10.1016/0167-2789(83)90112-4.
  2. Feigenbaum M. Universality in the behavior of nonlinear systems. Sov. Phys. Usp. 1983;141(2):343–374. DOI: 10.3367/UFNr.0141.198310e.0343.
  3. Vul EB, Sinai YG, Khanin KM. Feigenbaum universality and the thermodynamic formalism. Russian Math. Surveys. 1984;39(3):1–40. DOI: 10.1070/RM1984v039n03ABEH003162.
  4. Neimark YI, Landa PS. Stochastic and Chaotic Oscillations. Dordrecht: Springer; 1992. 500 p. DOI: 10.1007/978-94-011-2596-3.
  5. Kuznetsov SP. Dynamic Chaos. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  6. Greene JM, MacKay RS, Vivaldi F, and Feigenbaum MJ. Universal behavior in families of area-preserving maps. Physica D. 1981;3(3):468–486. DOI: 10.1016/0167-2789(81)90034-8.
  7. Collet P, Eckmann JP, and Koch H. On universality for area-preserving maps of the plane. Physica D. 1981;3(3):457–467. DOI: 10.1016/0167-2789(81)90033-6.
  8. Widom M, Kadanoff LP. Renormalization group analysis of bifurcations in area-preserving maps. Physica D. 1982;5(2–3):287–292. DOI: 10.1016/0167-2789(82)90023-9.
  9. Hu B, Rudnick J. Exact solution of the Feigenbaum renormalization group equations for intermittency. Phys. Rev. Lett. 1982;48(24):1645–1648. DOI: 10.1103/PhysRevLett.48.1645.
  10. Feigenbaum MJ, Kadanoff LP, and 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.
  11. Ostlund S, Rand D, Sethna J, and Siggia ED. Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. Physica D. 1983;8(3):303–342. DOI: 10.1016/0167-2789(83)90229-4.
  12. Collet P, Coullet P, and Tresser C. Scenarios under constraint. J. Physique Lett. 1985;46(4):L143–L147. DOI: 10.1051/jphyslet:01985004604014300.
  13. Greene JM, Mao J. Higher-order fixed points of the renormalisation operator for invariant circles. Nonlinearity. 1990;3(1):69–78. DOI: 10.1088/0951-7715/3/1/005.
  14. Kuznetsov AP, Kuznetsov SP, Sataev IR. Co-dimensionality and typicality in the context of the problem of describing the transition to chaos through period doubling in dissipative dynamical systems. Regular and Chaotic Dynamics. 1997;2(3–4)90–105 (in Russian).
  15. Kuznetsov AP, Kuznetsov SP, and Sataev IR. A variety of period-doubling universality classes in multi-parameter analysis of transition to chaos. Physica D. 1997;109(1–2):91–112. DOI: 10.1016/S0167-2789(97)00162-0.
  16. Kuznetsov AP, Kuznetsov SP, Sataev IR. Critical behavior in the transition to chaos through doubling the period. Model mappings and renormalization group analysis. In: Gaponov-Grekhov AV, Nekorkin VI, editors. Nonlinear Waves-2002. Nizhni Novgorod: IAP RAS; 2003. P. 395–415 (in Russian).
  17. Wilson K, Kohut J. Renormalization Group and ε-Decomposition. Moscow: Mir; 1975. 255 p. (in Russian).
  18. Balescu R. Equilibrium and Nonequilibrium Statistical Mechanics. Wiley-Blackwell; 1975. 756 p.
  19. Kuznetsov AP, Kuznetsov SP, and Sataev IR. Period doubling system under fractal signal: Bifurcation in the renormalization group equation. Chaos, Solitons and Fractals. 1991;1(4):355–367. DOI: 10.1016/0960-0779(91)90026-6.
  20. Kuznetsov AP, Kuznetsov SP, Sataev IR. Fractal signal and dynamics of systems demonstrating period doubling. Izvestiya VUZ. Applied Nonlinear Dynamics. 1995;3(5):64–87 (in Russian).
  21. Kuznetsov SP, Sataev IR. New types of critical dynamics for two-dimensional maps. Phys. Lett. A. 1992;162(3):236–242. DOI: 10.1016/0375-9601(92)90440-W.
  22. Kuznetsov SP, Sataev IR. Period-doubling for two-dimensional non-invertible maps: Renormalization group analysis and quantitative universality. Physica D. 1997;101(3–4):249–269. DOI: 10.1016/S0167-2789(96)00237-0.
  23. Hu B, Mao JM. Period doubling: Universality and critical-point order. Phys. Rev. A. 1982;25(6):3259–3261. DOI: 10.1103/PhysRevA.25.3259.
  24. Hu B, Satija II. A spectrum of universality classes in period doubling and period tripling. Phys. Lett. A. 1983;98(4):143–146. DOI: 10.1016/0375-9601(83)90569-8.
  25. Hauser PR, Tsallis C, and Curado EMF. Criticality of rouds to chaos of the 1 − a|x| z. Phys. Rev. A. 1984;30(4):2074–2079. DOI: 10.1103/PhysRevA.30.2074.
  26. Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer; 1983. 462 p. DOI: 10.1007/978-1-4612-1140-2.
  27. Kuznetsov SP, Sataev IR. Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillator. Phys. Rev. E. 2001;64(4):046214. DOI: 10.1103/PhysRevE.64.046214.
  28. Kuznetsov AP, Turukina LV, Savin AV, Sataev IR, Sedova JV, and Milovanov SV. Multi-parameter picture of transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics. 2002;10(3):80–96 (in Russian). 
Received: 
02.06.2005
Accepted: 
02.06.2005
Published: 
31.10.2005
Short text (in English):
PDF icon a2005no3p027.doc_.pdf
(downloads: 62)
  • 1653 reads