scholarly journals Logistic map trajectory distributions: Renormalization-group, entropy, and criticality at the transition to chaos

2021 ◽  
Vol 31 (3) ◽  
pp. 033112
Author(s):  
A. Diaz-Ruelas ◽  
F. Baldovin ◽  
A. Robledo
Keyword(s):  
Renormalization Group ◽  
Logistic Map ◽  
Transition To Chaos

scholarly journals Universality and scaling in networks of period-doubling maps with a pacemaker

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
Anna S. Ivanova ◽  
Sergey P. Kuznetsov ◽  
Andrew H. Osbaldestin
Keyword(s):  
Renormalization Group ◽  
Period Doubling ◽  
Critical Behaviour ◽  
Transition To Chaos ◽  
Single Period ◽  
Coupled Cells ◽  
Global Coupling ◽  
Coupled Maps ◽  
Simultaneous Transition

The networks of globally coupled maps with a pacemaker have been introduced. We consider a generalization of the Kaneko model with a pacemaker represented by a single period-doubling element coupled unidirectionally with a set of other mutually coupled cells. We also investigate the dynamics of a system of two unidirectionally coupled elements, which manifests a special type of critical behaviour, known asbicriticality, at the point of simultaneous transition to chaos in both subsystems. With the help of the renormalization group (RG), we show for a case of two mutually coupled bicritical maps with a pacemaker that there are two types of coupling: dissipative and inertial. We investigate the dynamics of a network with a pacemaker with two types of global coupling and the properties of universality and scaling in this system.

scholarly journals LYAPUNOV EXPONENTS FOR THE INTERMITTENT TRANSITION TO CHAOS

1999 ◽  
Vol 09 (04) ◽  
pp. 657-670
Author(s):  
JAMES HANSSEN ◽  
WALTER WILCOX
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Single Channel ◽  
Logistic Map ◽  
Transition To Chaos ◽  
Tangent Bifurcation ◽  
Discrete Nature ◽  
Improved Model ◽  
Modified Theory ◽  
Full Simulation

The dependence of the Lyapunov exponent on the closeness parameter, ε, in tangent bifurcation systems is investigated. We study and illustrate two averaging procedures for defining Lyapunov exponents in such systems. First, we develop theoretical expressions for an isolated tangency channel in which the Lyapunov exponent is defined on single channel passes. Numerical simulations were done to compare theory to measurement across a range of ε values. Next, as an illustration of defining the Lyapunov exponent on many channel passes, a simulation of the intermittent transition in the logistic map is described. The modified theory for the channels is explained and a simple model for the gate entrance rates is constructed. An important correction due to the discrete nature of the iterative flow is identified and incorporated in an improved model. Realistic fits to the data were made for the Lyapunov exponents from the logistic gate and from the full simulation. A number of additional corrections which could improve the treatment of the gates are identified and briefly discussed.

SELF-SIMILARITY AND UNIVERSALITY IN CHUA'S CIRCUIT VIA THE APPROXIMATE CHUA'S 1-D MAP

1993 ◽  
Vol 03 (02) ◽  
pp. 431-440 ◽  
Author(s):  
A. P. KUZNETSOV ◽  
S. P. KUZNETSOV ◽  
I. R. SATAEV ◽  
L. O. CHUA
Keyword(s):  
Renormalization Group ◽  
Critical Points ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Chua’S Circuit ◽  
Self Similarity ◽  
One Dimensional ◽  
Chua's Circuit ◽  
Transition To Chaos ◽  
Self Similar

In this paper we investigate the features of the transition to chaos in a one-dimensional Chua's map which describes approximately the Chua's circuit. These features arise from the nonunimodality of this map. We show that there exists a variety of types of critical points, which are characterized by a universal self-similar topography in a neighborhood of each critical point in the parameter plane. Such universalities are associated with various cycles of Feigenbaum's renormalization group equation.

scholarly journals DYNAMICS OF TWO LOGISTIC MAPS WITH A MULTIPLICATIVE COUPLING

1992 ◽  
Vol 02 (02) ◽  
pp. 421-425 ◽  
Author(s):  
R. LÓPEZ-RUIZ ◽  
C. PÉREZ-GARCÍA
Keyword(s):  
Dynamical Behavior ◽  
Logistic Map ◽  
Transition To Chaos ◽  
Route To Chaos

The dynamical behavior of a system formed by two symmetrically coupled logistic maps with a multiplicative coupling is analyzed. The transition to chaos (Ruelle-Takens type) and the multifractal properties of the attractor have been determined in this system. This transition cannot be deduced from the subharmonic cascade typical of a single logistic map. Under the same kind of symmetry, different classes of coupling in 2D maps give the same qualitative route to chaos but with different geometrical transition mechanisms.

RENORMALIZATION GROUP APPROACH TO THE EFFECT OF NOISE AT THE BIRTH OF A STRANGE NONCHAOTIC ATTRACTOR

2007 ◽  
Vol 07 (03) ◽  
pp. L263-L271
Author(s):  
SERGEY P. KUZNETSOV ◽  
IGOR R. SATAEV
Keyword(s):  
Renormalization Group ◽  
Additive Noise ◽  
Scaling Law ◽  
Period Doubling ◽  
Characteristic Time Scale ◽  
Group Approach ◽  
Strange Nonchaotic Attractor ◽  
Transition To Chaos ◽  
Renormalization Group Approach ◽  
Autonomous Differential Equations

Scaling regularities associated with additive noise are examined in a model of the pitch-fork bifurcation map with multiplicative quasiperiodic driving (Grebogi et al., Physica D13, 261) with the golden-mean frequency ratio at the birth of a strange nonchaotic attractor (SNA). This case of the onset of SNA termed as the blowout bifurcation route was discussed in the context of realistic systems governed by non-autonomous differential equations (Yalçynkaya and Lai, Phys. Rev. Lett., 77, 5039). Our method taking into the account of noise is based on renormalization group (RG) analysis of the birth of SNA (Kuznetsov et al., Phys. Rev. E51, 1629) with application of an appropriate generalization of the approach of Crutchfield et al. (Phys. Rev. Lett., 46, 933) and Shraiman et al. (Phys. Rev. Lett., 46, 935) originally developed for the period doubling transition to chaos. A constant γ=7.4246 is evaluated that determines the scaling law regarding the intensity of noise: A decrease of the noise amplitude by this factor allows resolving one more level of the fractal-like structure associated with the characteristic time scale which is increased by a factor of [Formula: see text]. Numeric results demonstrating evidence of the expected regularities are presented, e.g. portraits of the noisy attractors in different scales.

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