scholarly journals Superconvergence of topological entropy in the symbolic dynamics of substitution sequences

2019 ◽  
Vol 7 (2) ◽  
Author(s):  
Leon Zaporski ◽  
Felix Flicker
Keyword(s):  
Topological Entropy ◽  
Symbolic Dynamics ◽  
Convergence Rates ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Universality Class ◽  
Accumulation Point ◽  
Nonlinear Dynamical ◽  
Double Exponential ◽  
Symbolic Sequences

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.

Linear and Nonlinear Dynamic Behaviour

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Dynamic Behavior ◽  
Dynamic Systems ◽  
Dynamic Behaviour ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Nonlinear Dynamical ◽  
Deterministic Dynamic ◽  
Random Behavior ◽  
Single Solution ◽  
Equivalent Systems

In this chapter, we describe how highly erratic dynamic behavior can arise from a nonlinear logistic map, and how this apparently random behavior is governed by a surprising order. With this lesson in mind, we should not be overly surprised that highly erratic and random appearing observed data might also be generated by parsimonious deterministic dynamic systems. At a minimum, we contend that researchers should apply NLTS to test for this possibility. We also introduced tools to analyze dynamic behavior that form the foundation for NLTS. In particular, we have stressed the quite unexpected capability to achieve some form of predictability even with only one trajectory at hand. In subsequent chapters, we treat known nonlinear dynamical systems as unknown, and investigate how NLTS methods rely on a single solution (or multiple solutions) generated by them to reconstruct equivalent systems. This is a conventional approach in the literature for seeing how NLTS methods work since we know what needs to be reconstructed.

scholarly journals Computation of the topological entropy in chaotic biophysical bursting models for excitable cells

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Jorge Duarte ◽  
Luís Silva ◽  
J. Sousa Ramos
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Excitable Cells ◽  
Physiological Models ◽  
Nonlinear Dynamical ◽  
Chaotic Bursting ◽  
Oscillatory State ◽  
One Dimensional Maps

One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described. In the present paper, we study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics on the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

scholarly journals NOMINAL CROSS RECURRENCE AS A GENERALIZED LAG SEQUENTIAL ANALYSIS FOR BEHAVIORAL STREAMS

2011 ◽  
Vol 21 (04) ◽  
pp. 1153-1161 ◽  
Author(s):  
RICK DALE ◽  
ANNE S. WARLAUMONT ◽  
DANIEL C. RICHARDSON
Keyword(s):  
Sequential Analysis ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Coarse Graining ◽  
Human Interaction ◽  
Coarse Grained ◽  
Nonlinear Dynamical ◽  
Movement Data ◽  
Lag Sequential Analysis ◽  
Quantification Analysis

We briefly present lag sequential analysis for behavioral streams, a commonly used method in psychology for quantifying the relationships between two nominal time series. Cross recurrence quantification analysis (CRQA) is shown as an extension of this technique, and we exemplify this nominal application of CRQA to eye-movement data in human interaction. In addition, we demonstrate nominal CRQA in a simple coupled logistic map simulation used in previous communication research, permitting the investigation of properties of nonlinear systems such as bifurcation and onset to chaos, even in the streams obtained by coarse-graining a coupled nonlinear model. We end with a summary of the importance of CRQA for exploring the relationship between two behavioral streams, and review a recent theoretical trend in the cognitive sciences that would be usefully informed by this and similar nonlinear methods. We hope this work will encourage scientists interested in general properties of complex, nonlinear dynamical systems to apply emerging methods to coarse-grained, nominal units of measure, as there is an immediate need for their application in the psychological domain.

scholarly journals STUDY OF BIFURCATION BEHAVIOR OF LDPC DECODERS

2006 ◽  
Vol 16 (11) ◽  
pp. 3435-3449 ◽  
Author(s):  
XIA ZHENG ◽  
FRANCIS C. M. LAU ◽  
CHI K. TSE ◽  
S. C. WONG
Keyword(s):  
Finite Length ◽  
Convergence Rates ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Ldpc Codes ◽  
Error Performance ◽  
Signal To Noise ◽  
Nonlinear Dynamical ◽  
Ldpc Decoder ◽  
Bifurcation Phenomena

The use of low-density-parity-check (LDPC) codes in coding digital messages has aroused much research interest because of their excellent bit-error performance. The behavior of the iterative LDPC decoders of finite length, however, has not been fully evaluated under different signal-to-noise conditions. By considering the finite-length LDPC decoders as high-dimensional nonlinear dynamical systems, we attempt to investigate their dynamical behavior and bifurcation phenomena for a range of signal-to-noise ratios (SNRs). Extensive simulations have been performed on both regular and irregular LDPC codes. Moreover, we derive the Jacobian of the system and calculate the corresponding eigenvalues. Results show that bifurcations, including fold, flip and Neimark–Sacker bifurcations, are exhibited by the LDPC decoder. Results are useful for optimizing the choice of parameters that may enhance the effectiveness of the decoding algorithm and improve the convergence rates.

Ljapunov Exponents, Hyperchaos and Hurst Exponent

2005 ◽  
Vol 60 (4) ◽  
pp. 252-254 ◽  
Author(s):  
Willi-Hans Steeb ◽  
Eugenio Cosme Andrieu
Keyword(s):  
Dynamical Systems ◽  
Hurst Exponent ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Two Dimensional ◽  
Nonlinear Dynamical

Abstract We consider nonlinear dynamical systems with chaotic and hyperchaotic behaviour.We investigate the behaviour of the Hurst exponent at the transition from chaos to hyperchaos. A two-dimensional coupled logistic map is studied.

scholarly journals Symbolic dynamics, entropy and complexity of the feigenbaum map at the accumulation point

1998 ◽  
Vol 2 (3) ◽  
pp. 187-194 ◽  
Author(s):  
Werner Ebelings ◽  
Katja Rateitschak
Keyword(s):  
Complex System ◽  
Long Range ◽  
Symbolic Dynamics ◽  
Logistic Map ◽  
Accumulation Point ◽  
Long Range Correlations ◽  
Complexity Function

This paper aims to make further contributions to the exploration of the symbolic dynamics generated by the logistic map at Feigenbaum accumulation point. In particular we are interested in the grammar of these sequences; completing earlier studies we study here arbitrary partitions also. Our main aim is the investigation of the special grammars which characterize the long-range correlations between letters. Considering these sequences as standard examples of a complex system, we introduce and discuss a complexity function derived from the conditional entropies. Further we discuss local predictabilities.

scholarly journals Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

2018 ◽  
Author(s):  
Geoff Boeing
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Models ◽  
Logistic Map ◽  
Three Dimensional ◽  
Basins Of Attraction ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
One Dimensional Maps

Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

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