BIFURCATION OF UNIVERSAL REGIMES AT THE BORDER OF CHAOS
Cite this article as:
Kuznetsov S. P., Mailybaev А. А., 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: https://doi.org/10.18500/0869-6632-2005-13-3-27-47
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 k 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 k. At k = 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 k from the critical value kc = 1; 984396. The approach is analogous to that known in the phase transition theory as "-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.
1. Feigenbaum M.J. Universal behavior in nonlinear systems // Physica D. 1983. Vol. 7. P. 16–39.
2. Фейгенбаум М. Универсальность в поведении нелинейных систем // УФН. 1983. Т. 141, No 2. С. 343–374.
3. Вул Е.Б., Синай Я.Г., Ханин К.М. Универсальность Фейгенбаума и термодинамический формализм // УМН. 1984. Т. 39, No 3. С. 3–37.
4. Неймарк Ю.И., Ланда П.С. Стохастические и хаотические колебания. М.: Наука, 1987. 424 с.
5. Кузнецов С.П. Динамический хаос. М.: Физматлит, 2001. 296 c.
6. Greene J.M., MacKay R.S., Vivaldi F., and Feigenbaum M.J. Universal behavior in families of area preserving maps // Physica D. 1981. Vol. 3. P. 468–486.
7. Collet P., Eckmann J.-P., and Koch H. On universality for area-preserving maps of the plane // Physica D. 1981. Vol. 3. P. 457–467.
8. Widom M. and Kadanoff L.P. Renormalization group analysis of area-preserving maps // Physica D. 1982. Vol. 5. P. 287–292.
9. Hu B. and Rudnick J. Exact solution of the Feigenbaum renormalization group equations for intermittency // Phys. Rev. Lett. 1982. Vol. 48. P. 1645–1648.
10. Feigenbaum M.J., Kadanoff L.P., and Shenker S.J. Quasiperiodicity in dissipative systems: A renormalization group analysis // Physica D. 1982. Vol. 5. P. 370–386.
11. Ostlund S., Rand D., Sethna J., and Siggia E.D. Universal properties of the transition from quasi-periodicity to chaos in dissipative systems // Physica D. 1983. Vol. 8. P. 303–342.
12. Collet P., Coullet P., and Tresser C. Scenarios under constraint// J. Physique Lett. 1985. Vol. 46. P. L143–L147.
13. Greene J.M. and Mao J. Higher-order fixed points of the renormalisation operator for invariant circles // Nonlinearity. 1990. Vol. 3. P. 69–78.
14. Кузнецов А.П., Кузнецов С.П., Сатаев И.Р. Коразмерность и типичность в контексте проблемы описания перехода к хаосу через удвоения периода в диссипативных динамических системах // Регулярная и хаотическая динамика. 1997. Т. 2, No3–4. C. 90–105.
15. Kuznetsov A.P., Kuznetsov S.P., and Sataev I.R. A variety of period-doubling universality classes in multi-parameter analysis of transition to chaos // Physica D. 1997. Vol. 109. P. 91–112.
16. Кузнецов А.П., Кузнецов С.П., Сатаев И.Р. Критическое поведение при переходе к хаосу через удвоения периода. Модельные отображения и ренормгрупповой анализ // В кн.: Нелинейные волны 2002 / Под ред. А.В.Гапонова-Грехова и В.И.Некоркина. Н.Новгород: ИПФ РАН, 2003. C. 395–415.
17. Вильсон К., Когут Дж. Ренормализационная группа и ε-разложение. М.: Мир, 1975. 255 с.
18. Балеску Р. Равновесная и неравновесная статистическая механика. М.: Мир, 1978. Т. 1. 405 с.
19. Kuznetsov A.P., Kuznetsov S.P. and Sataev I.R. Period doubling system under fractal signal: Bifurcation in the renormalization group equation // Chaos, Solitons and Fractals. 1991. Vol. 1. P. 355–367.
20. Кузнецов А.П., Кузнецов С.П., Сатаев И.Р. Фрактальный сигнал и динамика систем, демонстрирующих удвоения периода // Изв. вузов. Прикладная нелинейная динамика. 1995. Т. 3, No5. C. 64–87.
21. Kuznetsov S.P. and Sataev I.R. New types of critical dynamics for two-dimensional maps // Phys. Lett. A. 1992. Vol. 162. P. 236–242.
22. Kuznetsov S.P. and Sataev I.R. Period-doubling for two-dimensional non-invertible maps: Renormalization group analysis and quantitative universality // Physica D. 1997. Vol. 101. P. 249–269.
23. Hu B. and Mao J.M. Period doubling: Universality and critical-point order // Phys. Rev. A. 1982. Vol. 25. P. 3259–3261.
24. Hu B. and Satija I.I. A spectrum of universality classes in period-doubling and period tripling // Phys. Lett. A. 1983. Vol. 98. P. 143–146.
25. Hauser P.R., Tsallis C., and Curado E.M.F. Criticality of rouds to chaos of the 1 − a|x|
26. Guckenheimer J. and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York, Springer, 1983.
27. Kuznetsov S.P. and Sataev I.R. Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rossler oscillator // Phys. Rev. E. 2001. Vol. 64. P. 046214.
28. Kuznetsov A.P., Turukina L.V., Savin A.V., Sataev I.R., Sedova J.V., and Milovanov S.V. Multi-parameter picture of transition to chaos // Izvestiya vuz. Applied Nonlinear Dynamics (Saratov). 2002. Vol. 10, No 3. P. 80–96.
BibTeX
author = {Sergey P. Kuznetsov and А. А. Mailybaev and I. R. Sataev},
title = {BIFURCATION OF UNIVERSAL REGIMES AT THE BORDER OF CHAOS},
year = {2005},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {13},number = {3},
url = {https://old-andjournal.sgu.ru/en/articles/bifurcation-of-universal-regimes-at-the-border-of-chaos},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2005-13-3-27-47},pages = {27--47},issn = {0869-6632},
keywords = {-},
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 k 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 k. At k = 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 k from the critical value kc = 1; 984396. The approach is analogous to that known in the phase transition theory as "-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. }}