Special Issue "Chaos and Randomness of Discrete Dynamical Systems: Their Use in Applied Science"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 January 2018).

Special Issue Editor

Prof. Dr. Christophe Guyeux
E-Mail Website
Guest Editor
Femto-ST Institute, Université de Bourgogne Franche-Comté, Besançon, France
Interests: chaos; cryptography; discrete mathematics; bioinformatics

Special Issue Information

Dear Colleagues,

Chaotic behaviors of discrete dynamical systems have been extensively studied in the last five decades. From a theoretical perspective, dynamical systems strongly rely on mathematical topology and measure theory. Such notions as topological mixing, ergodicity, Lyapunov exponents, etc., are core tools in the field. Graph theory and the Markov chain formalism have also played a key role in understanding randomness, leading to a trustworthy analysis of the statistical properties of discrete dynamical systems. These theoretical breakthroughs have been historically applied to the study of complex phenomena coming from, e.g., physics and biology, where qualitative and quantitative analysis of chaotic dynamics have been completed to some extent. More recently, new unforeseen applications of discrete dynamical systems and their complexity have emerged, in particular for numerical simulations, new computational principles (chaos computing), and information security (chaotic pseudorandom number generators). Conversely, recent advances in big data, bioinformatics, and deep learning, have revealed new types of complex dynamics, which can be, at least partially, understood or simplified by means of discrete dynamical system modeling.

The aim of this Special Issue is to bring together both mathematical theorists and applied scientists working on chaotic dynamics or random dynamics in discrete systems. New theoretical results in the fields of mathematical chaos, Markov chains, or discrete dynamical systems are welcome, together with applications related to their randomness or unpredictable behaviors.

Prof. Dr. Christophe Guyeux
Guest Editor

Manuscript Submission Information

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Keywords

  • discrete dynamical systems
  • chaos
  • Markov chains
  • randomness
  • applied science.

Published Papers (6 papers)

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Research

Open AccessArticle
Decomposition of Dynamical Signals into Jumps, Oscillatory Patterns, and Possible Outliers
Mathematics 2018, 6(7), 124; https://doi.org/10.3390/math6070124 - 16 Jul 2018
Abstract
In this note, we present a component-wise algorithm combining several recent ideas from signal processing for simultaneous piecewise constants trend, seasonality, outliers, and noise decomposition of dynamical time series. Our approach is entirely based on convex optimisation, and our decomposition is guaranteed to [...] Read more.
In this note, we present a component-wise algorithm combining several recent ideas from signal processing for simultaneous piecewise constants trend, seasonality, outliers, and noise decomposition of dynamical time series. Our approach is entirely based on convex optimisation, and our decomposition is guaranteed to be a global optimiser. We demonstrate the efficiency of the approach via simulations results and real data analysis. Full article
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Open AccessFeature PaperArticle
Gray Codes Generation Algorithm and Theoretical Evaluation of Random Walks in N-Cubes
Mathematics 2018, 6(6), 98; https://doi.org/10.3390/math6060098 - 08 Jun 2018
Abstract
In previous works, some of the authors have proposed a canonical form of Gray Codes (GCs) in N-cubes (hypercubes of dimension N). This form allowed them to draw an algorithm that theoretically provides exactly all the GCs for a given dimension N [...] Read more.
In previous works, some of the authors have proposed a canonical form of Gray Codes (GCs) in N-cubes (hypercubes of dimension N). This form allowed them to draw an algorithm that theoretically provides exactly all the GCs for a given dimension N. In another work, we first have shown that any of these GC can be used to build the transition function of a Pseudorandom Number Generator (PRNG). Also, we have found a theoretical quadratic upper bound of the mixing time, i.e., the number of iterations that are required to provide a PRNG whose output is uniform. This article, extends these two previous works both practically and theoretically. On the one hand, another algorithm for generating GCs is proposed that provides an efficient generation of subsets of the entire set of GCs related to a given dimension N. This offers a large choice of GC to be used in the construction of Choatic Iterations based PRNGs (CI-PRNGs), leading to a large class of possible PRNGs. On the other hand, the mixing time has been theoretically shown to be in Nlog(N), which was anticipated in the previous article, but not proven. Full article
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Open AccessArticle
Theoretical Study of the One Self-Regulating Gene in the Modified Wagner Model
Mathematics 2018, 6(4), 58; https://doi.org/10.3390/math6040058 - 09 Apr 2018
Abstract
Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, [...] Read more.
Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration of how to apply such investigations in the case of the dynamical system representing the one self-regulating gene. In this context, we focused on the composite function f a ( m . x ) where f a is the parametric sigmoid function and m is a scalar not in { 0 , 1 } and we have proven that the number of fixed-point can be deduced theoretically, according to the values of a and m. Full article
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Open AccessArticle
Asynchronous Iterations of Parareal Algorithm for Option Pricing Models
Mathematics 2018, 6(4), 45; https://doi.org/10.3390/math6040045 - 21 Mar 2018
Cited by 4
Abstract
Spatial domain decomposition methods have been largely investigated in the last decades, while time domain decomposition seems to be contrary to intuition and so is not as popular as the former. However, many attractive methods have been proposed, especially the parareal algorithm, which [...] Read more.
Spatial domain decomposition methods have been largely investigated in the last decades, while time domain decomposition seems to be contrary to intuition and so is not as popular as the former. However, many attractive methods have been proposed, especially the parareal algorithm, which showed both theoretical and experimental efficiency in the context of parallel computing. In this paper, we present an original model of asynchronous variant based on the parareal scheme, applied to the European option pricing problem. Some numerical experiments are given to illustrate the convergence performance and computational efficiency of such a method. Full article
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Open AccessArticle
Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models
Mathematics 2018, 6(3), 39; https://doi.org/10.3390/math6030039 - 07 Mar 2018
Abstract
We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two [...] Read more.
We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend a relatively long time visiting the attractor’s ruins. Full article
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Open AccessFeature PaperArticle
A Simple Formula for the Hilbert Metric with Respect to a Sub-Gaussian Cone
Mathematics 2018, 6(3), 35; https://doi.org/10.3390/math6030035 - 02 Mar 2018
Abstract
The Hilbert metric is a widely used tool for analysing the convergence of Markov processes and the ergodic properties of deterministic dynamical systems. A useful representation formula for the Hilbert metric was given by Liverani. The goal of the present paper is to [...] Read more.
The Hilbert metric is a widely used tool for analysing the convergence of Markov processes and the ergodic properties of deterministic dynamical systems. A useful representation formula for the Hilbert metric was given by Liverani. The goal of the present paper is to extend this formula to the non-compact and multidimensional setting with a different cone, taylored for sub-Gaussian tails. Full article
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