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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:


Bifurcation of universal regimes at the border of chaos

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

It is shown that a ?xed 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 ?rst 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 de?ection 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» ?xed point of renormalization group transformation to a new ?xed point, responsible for critical behavior with nontrivial critical indices.

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