Journal of Statistical Physics

, Volume 130, Issue 3, pp 599–616 | Cite as

Birth of a New Class of Period-Doubling Scaling Behavior as a Result of Bifurcation in the Renormalization Equation

  • S. P. Kuznetsov
  • A. A. Mailybaev
  • I. R. Sataev


It is found that a fixed point of the renormalization group equation corresponding to a system of a unimodal map with extremum of power κ and a map summarizing values of a function of the dynamical variable of the first subsystem, undergoes a bifurcation in the course of increase of κ. It occurs at κc=1.984396 and results in a birth of the period-2 stationary solution of the RG equation. At κ=2 this period-2 solution corresponds to the universal period-doubling behavior discovered earlier and denoted as the C-type criticality (Kuznetsov and Sataev in Phys. Lett. A 162:236–242, 1992). By combination of analytical methods and numerical computations we obtain and analyze an asymptotic expansion of the period-2 solution in powers of Δκ=κκc. The developed approach resembles the ε-expansion in the phase transition theory, in which a “trivial” stationary point of the RG transformation undergoes a bifurcation that gives rise to a new fixed point responsible for the critical behavior with nontrivial critical indices.


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • S. P. Kuznetsov
    • 1
  • A. A. Mailybaev
    • 2
  • I. R. Sataev
    • 1
  1. 1.Institute of Radio-Engineering and Electronics of RASSaratov BranchSaratovRussia
  2. 2.Institute of MechanicsMoscow State Lomonosov UniversityMoscowRussia

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