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

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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

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Authors and Affiliations

  1. Institute of Radio-Engineering and Electronics of RAS, Zelenaya 38, Saratov, 410019, Russian Federation

    S. P. Kuznetsov, A. P. Kuznetsov & I. R. Sataev

  2. Faculty of Nonlinear Processes, Saratov State University, Bolshaya Kazachya, 112A, Saratov, 410012, Russian Federation

    S. P. Kuznetsov & A. P. Kuznetsov

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  1. S. P. Kuznetsov
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The paper is dedicated to the 60th birthday of Mitchel J. Feigenbaum

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Kuznetsov, S.P., Kuznetsov, A.P. & Sataev, I.R. Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps. J Stat Phys 121, 697–748 (2005). https://doi.org/10.1007/s10955-005-6973-6

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