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Journal of Statistical Physics

, Volume 121, Issue 5–6, pp 697–748 | Cite as

Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps

  • S. P. KuznetsovEmail author
  • A. P. Kuznetsov
  • I. R. Sataev
Article

Abstract

We review critical situations, linked with period-doubling transition to chaos, which require using at least two-dimensional maps as models representing the universality classes. Each of them corresponds to a saddle solution of the two-dimensional generalization of Feigenbaum-Cvitanović equation and is characterized by a set of distinct universal constants analogous to Feigenbaum’s α and δ. One type of criticality designated H was discovered by several authors in 80-th in the context of period doubling in conservative dynamics, but occurs as well in dissipative dynamics, as a phenomenon of codimension 2. Second is bicritical behavior, which takes place in systems allowing decomposition onto two dissipative period-doubling subsystems, each of which is brought by parameter tuning onto a threshold of chaos. Types of criticality designated as FQ and C occur in non-invertible two-dimensional maps. We present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems. In particular, we indicate a possibility for realization of the H type criticality without vanishing dissipation, but with its compensation in a self-oscillatory system. Next, we present a number of examples (coupled Hénon-like maps, coupled driven oscillators, coupled chaotic self-oscillators), which manifest bicritical behavior. For FQ-type we indicate possibility to arrange it in non-symmetric systems of coupled period-doubling subsystems, e.g. in Hénon-like maps and in Chua’s circuits. For C-type we present examples of its appearance in a driven Rössler oscillator at the period-doubling accumulation on the edge of syncronization tongue and in a model map with the Neimark–Sacker bifurcation

Keywords

Period-doubling onset of chaos renormalization group universality scaling coupled maps coupled oscillators multi-parameter analysis 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. P. Kuznetsov
    • 1
    • 2
    Email author
  • A. P. Kuznetsov
    • 1
    • 2
  • I. R. Sataev
    • 1
  1. 1.Institute of Radio-Engineering and Electronics of RASSaratovRussian Federation
  2. 2.Faculty of Nonlinear ProcessesSaratov State UniversitySaratovRussian Federation

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