scholarly journals SYMBOL-TO-SYMBOL CORRELATION FUNCTION AT THE FEIGENBAUM POINT OF THE LOGISTIC MAP

2013 ◽  
Vol 23 (07) ◽  
pp. 1350118 ◽  
Author(s):  
K. KARAMANOS ◽  
I. S. MISTAKIDIS ◽  
S. I. MISTAKIDIS
Keyword(s):  
Correlation Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Correlation Functions ◽  
Symbolic Dynamics ◽  
Numerical Experiments ◽  
Statistical Physics ◽  
Logistic Map ◽  
Algorithmic Structure ◽  
Significant Attention

Recently, simple dynamical systems such as the 1-d maps on the interval, gained significant attention in the context of statistical physics and complex systems. The decay of correlations in these systems, can be characterized and measured by correlation functions. In the context of symbolic dynamics of the nonchaotic multifractal attractors (i.e. Feigenbaum attractors), one observable, the symbol-to-symbol correlation function, for the generating partition of the logistic map, is rigorously introduced and checked by numerical experiments. Thanks to the Metropolis–Stein–Stein (MSS) algorithm, this observable can be calculated analytically, giving predictions in absolute accordance with numerical computations. The deep, algorithmic structure of the observable is revealed clearly reflecting the complexity of the multifractal attractor.

SYMBOLIC DYNAMICS, COARSE GRAINING AND THE MONITORING OF COMPLEX SYSTEMS

2011 ◽  
Vol 21 (12) ◽  
pp. 3465-3475 ◽  
Author(s):  
VASILEIOS BASIOS ◽  
DÓNAL MAC KERNAN
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Complex Systems ◽  
Error Estimates ◽  
Time Scales ◽  
Symbolic Dynamics ◽  
Prediction Error ◽  
Probabilistic Approach ◽  
Coarse Graining ◽  
Complex Nonlinear Systems

Coarse graining techniques and their associated symbolic dynamics are reviewed with a focus on probabilistic aspects of complex dynamical systems. The probabilistic approach initiated by Nicolis and coworkers has been elaborated. One of the major issues when dealing with the dynamics of complex nonlinear systems, the fact that the inherent time-scales of the unfolding phenomena are not well separated, is brought into focus. Recent results related to this interdependence, which is one of the most characteristic aspects of complexity and a major challenge in prediction, error estimates and monitoring of nonlinear complex systems, are discussed.

scholarly journals Various Auto-Correlation Functions of m-Bit Random Numbers Generated from Chaotic Binary Sequences

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1295
Author(s):  
Akio Tsuneda
Keyword(s):  
Correlation Function ◽  
Correlation Functions ◽  
Numerical Experiments ◽  
Binary Sequences ◽  
Random Numbers ◽  
One Dimensional ◽  
Auto Correlation ◽  
Auto Correlation Function ◽  
Nonlinear Maps ◽  
Correlation Properties

This paper discusses the auto-correlation functions of m-bit random numbers obtained from m chaotic binary sequences generated by one-dimensional nonlinear maps. First, we provide the theoretical auto-correlation function of an m-bit sequence obtained by m binary sequences that are assumed to be uncorrelated to each other. The auto-correlation function is expressed by a simple form using the auto-correlation functions of the binary sequences. This implies that the auto-correlation properties of the m-bit sequences can be easily controlled by the auto-correlation functions of the original binary sequences. In numerical experiments using a computer, we generated m-bit random sequences using some chaotic binary sequences with prescribed auto-correlations generated by one-dimensional chaotic maps. The numerical experiments show that the numerical auto-correlation values are almost equal to the corresponding theoretical ones, and we can generate m-bit sequences with a variety of auto-correlation properties. Furthermore, we also show that the distributions of the generated m-bit sequences are uniform if all of the original binary sequences are balanced (i.e., the probability of 1 (or 0) is equal to 1/2) and independent of one another.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals From correlation functions to event shapes in QCD

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
D. Chicherin ◽  
J. M. Henn ◽  
E. Sokatchev ◽  
K. Yan
Keyword(s):  
Correlation Function ◽  
Correlation Functions ◽  
Event Shape ◽  
Proof Of Concept ◽  
Yang Mills ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Event Shapes ◽  
A Charge ◽  
Conserved Currents

Abstract We present a method for calculating event shapes in QCD based on correlation functions of conserved currents. The method has been previously applied to the maximally supersymmetric Yang-Mills theory, but we demonstrate that supersymmetry is not essential. As a proof of concept, we consider the simplest example of a charge-charge correlation at one loop (leading order). We compute the correlation function of four electromagnetic currents and explain in detail the steps needed to extract the event shape from it. The result is compared to the standard amplitude calculation. The explicit four-point correlation function may also be of interest for the CFT community.

scholarly journals On the three-dimensional spatial correlations of curved dislocation systems

2021 ◽  
Vol 5 (1) ◽  
Author(s):  
Joseph Pierre Anderson ◽  
Anter El-Azab
Keyword(s):  
Correlation Function ◽  
Slip System ◽  
Correlation Functions ◽  
Mean Field ◽  
Coarse Graining ◽  
Dislocation Dynamics ◽  
Coarse Grained ◽  
Spatial Correlation Function ◽  
Line System ◽  
Dislocation Densities

AbstractCoarse-grained descriptions of dislocation motion in crystalline metals inherently represent a loss of information regarding dislocation-dislocation interactions. In the present work, we consider a coarse-graining framework capable of re-capturing these interactions by means of the dislocation-dislocation correlation functions. The framework depends on a convolution length to define slip-system-specific dislocation densities. Following a statistical definition of this coarse-graining process, we define a spatial correlation function which will allow the arrangement of the discrete line system at two points—and thus the strength of their interactions at short range—to be recaptured into a mean field description of dislocation dynamics. Through a statistical homogeneity argument, we present a method of evaluating this correlation function from discrete dislocation dynamics simulations. Finally, results of this evaluation are shown in the form of the correlation of dislocation densities on the same slip-system. These correlation functions are seen to depend weakly on plastic strain, and in turn, the dislocation density, but are seen to depend strongly on the convolution length. Implications of these correlation functions in regard to continuum dislocation dynamics as well as future directions of investigation are also discussed.

scholarly journals A journey from reductionist to systemic cell biology aboard the schooner Tara

2012 ◽  
Vol 23 (13) ◽  
pp. 2403-2406 ◽  
Author(s):  
Eric Karsenti
Keyword(s):  
Complex Systems ◽  
Molecular Interactions ◽  
Cell Biology ◽  
Statistical Physics ◽  
Interaction Networks ◽  
Cellular Functions ◽  
Personal Journey ◽  
New Approaches ◽  
The World ◽  
The Way

In this essay I describe my personal journey from reductionist to systems cell biology and describe how this in turn led to a 3-year sea voyage to explore complex ocean communities. In describing this journey, I hope to convey some important principles that I gleaned along the way. I realized that cellular functions emerge from multiple molecular interactions and that new approaches borrowed from statistical physics are required to understand the emergence of such complex systems. Then I wondered how such interaction networks developed during evolution. Because life first evolved in the oceans, it became a natural thing to start looking at the small organisms that compose the plankton in the world's oceans, of which 98% are … individual cells—hence the Tara Oceans voyage, which finished on 31 March 2012 in Lorient, France, after a 60,000-mile around-the-world journey that collected more than 30,000 samples from 153 sampling stations.

A Side-Peak Cancellation Scheme for Unambiguous BOC Signal Tracking

2015 ◽  
Vol 764-765 ◽  
pp. 462-465
Author(s):  
Keun Hong Chae ◽  
Hua Ping Liu ◽  
Seok Ho Yoon
Keyword(s):  
Correlation Function ◽  
Correlation Functions ◽  
Numerical Results ◽  
Tracking Performance ◽  
Side Peak ◽  
Signal Tracking ◽  
Partial Correlations

In this paper, we propose a side-peak cancellation scheme for unambiguous BOC signal tracking. We obtain partial correlations using a pulse model of a BOC signal, and then, we finally obtain an unambiguous correlation function based on the partial correlations. The proposed correlation function is confirmed from numerical results to provide an improved tracking performance over the conventional correlation functions.

A case study in bifurcation theory

2017 ◽  
Vol 28 (08) ◽  
pp. 1750104 ◽  
Author(s):  
Youssef Khmou
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Logistic Map ◽  
Logistic Function ◽  
Second Order ◽  
Short Paper ◽  
Chaotic Region

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.

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