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Entropy
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Nonlinear Dynamics: Bifurcations of Periodic and Chaotic Oscillations
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Nonlinear Dynamics: Bifurcations of Periodic and Chaotic Oscillations

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 8134

Special Issue Editors

Dr. Wieslaw Marszalek

E-Mail Website
Guest Editor
Department of Computer Science, Opole University of Technology, 45-758 Opole, Poland
Interests: nonlinear dynamics; time-series analysis; entropy-based analysis; bifurcations and chaos; machine learning; memristors; parallel computing
Special Issues, Collections and Topics in MDPI journals
Dr. Helmut Podhaisky

E-Mail Website
Guest Editor
Department of Mathematics, Martin-Luther University, Halle (Saale), Germany
Interests: numerical applied mathematics; software applications; machine learning

Special Issue Information

Dear Colleagues,

Nonlinear dynamical systems in engineering and science offer a wide spectrum of interesting phenomena that are typically investigated through numerical computations and advanced methods. When parameters of such systems vary, the systems’ responses bifurcate and change their nature. Oscillatory dynamical models, described by systems of nonlinear ordinary differential equations are of special interests, as they appear in engineering, physics, chemistry, biology, astronomy, economy and other areas.

In this Special Issue, we aim to publish new results in computational aspects of analyzing periodic and chaotic nonlinear dynamical systems by using entropy-based methods. We are particularly interested in papers describing new computational results in nonlinear oscillatory circuits and systems obtained through new entropy related methods in software applications, machine learning and parallel computing. Reports on new or improved numerical algorithms and tests for chaos applied to oscillatory dynamical models in engineering and science are also welcome.

Prof. Wieslaw Marszalek
Dr. Helmut Podhaisky
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • computations in oscillatory dynamics
  • bifurcation and chaos
  • oscillations in engineering and science
  • machine learning in oscillatory dynamical systems
  • software applications
  • parallel computation in nonlinear dynamics

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Published Papers (3 papers)

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Research

8 pages, 1047 KiB  
Article
On Physically Unacceptable Numerical Solutions Yielding Strong Chaotic Signals
by Wieslaw Marszalek
Entropy 2022, 24(6), 769; https://doi.org/10.3390/e24060769 - 30 May 2022
Viewed by 1936
Abstract
Physically unacceptable chaotic numerical solutions of nonlinear circuits and systems are discussed in this paper. First, as an introduction, a simple example of a wrong choice of a numerical solver to deal with a second-order linear ordinary differential equation is presented. Then, the [...] Read more.
Physically unacceptable chaotic numerical solutions of nonlinear circuits and systems are discussed in this paper. First, as an introduction, a simple example of a wrong choice of a numerical solver to deal with a second-order linear ordinary differential equation is presented. Then, the main result follows with the analysis of an ill-designed numerical approach to solve and analyze a particular nonlinear memristive circuit. The obtained trajectory of the numerical solution is unphysical (not acceptable), as it violates the presence of an invariant plane in the continuous systems. Such a poor outcome is then turned around, as we look at the unphysical numerical solution as a source of strong chaotic sequences. The 0–1 test for chaos and bifurcation diagrams are applied to prove that the unacceptable (from the continuous system point of view) numerical solutions are, in fact, useful chaotic sequences with possible applications in cryptography and the secure transmission of data. Full article
(This article belongs to the Special Issue Nonlinear Dynamics: Bifurcations of Periodic and Chaotic Oscillations)
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15 pages, 2884 KiB  
Article
Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams
by Wieslaw Marszalek, Jan Sadecki and Maciej Walczak
Entropy 2021, 23(7), 876; https://doi.org/10.3390/e23070876 - 8 Jul 2021
Cited by 2 | Viewed by 2662
Abstract
Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few [...] Read more.
Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation. Full article
(This article belongs to the Special Issue Nonlinear Dynamics: Bifurcations of Periodic and Chaotic Oscillations)
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15 pages, 3210 KiB  
Article
Enhancing Chaos Complexity of a Plasma Model through Power Input with Desirable Random Features
by Hayder Natiq, Muhammad Rezal Kamel Ariffin, Muhammad Asyraf Asbullah, Zahari Mahad and Mohammed Najah
Entropy 2021, 23(1), 48; https://doi.org/10.3390/e23010048 - 30 Dec 2020
Cited by 10 | Viewed by 2419
Abstract
The present work introduces an analysis framework to comprehend the dynamics of a 3D plasma model, which has been proposed to describe the pellet injection in tokamaks. The analysis of the system reveals the existence of a complex transition from transient chaos to [...] Read more.
The present work introduces an analysis framework to comprehend the dynamics of a 3D plasma model, which has been proposed to describe the pellet injection in tokamaks. The analysis of the system reveals the existence of a complex transition from transient chaos to steady periodic behavior. Additionally, without adding any kind of forcing term or controllers, we demonstrate that the system can be changed to become a multi-stable model by injecting more power input. In this regard, we observe that increasing the power input can fluctuate the numerical solution of the system from coexisting symmetric chaotic attractors to the coexistence of infinitely many quasi-periodic attractors. Besides that, complexity analyses based on Sample entropy are conducted, and they show that boosting power input spreads the trajectory to occupy a larger range in the phase space, thus enhancing the time series to be more complex and random. Therefore, our analysis could be important to further understand the dynamics of such models, and it can demonstrate the possibility of applying this system for generating pseudorandom sequences. Full article
(This article belongs to the Special Issue Nonlinear Dynamics: Bifurcations of Periodic and Chaotic Oscillations)
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