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Kuznecov 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


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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