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Symmetry
Special Issues
Three-Dimensional Dynamical Systems and Symmetry
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Three-Dimensional Dynamical Systems and Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 November 2024 | Viewed by 6990

Special Issue Editors

Dr. Cristian Lazureanu

E-Mail Website
Guest Editor
Department of Mathematics, Politehnica University of Timişoara, Piata Victoriei No. 2, 300006 Timişoara, Romania
Interests: Hamilton-Poisson systems; nonlinear dynamical systems; bifurcations; mathematical models
Special Issues, Collections and Topics in MDPI journals
Dr. Dana Constantinescu

E-Mail Website
Guest Editor
Department of Applied Mathematics, University of Craiova, 200585 Craiova, Romania
Interests: mathematics; physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Dynamical systems are used to model some processes from various fields, and they are widely investigated from many points of view. The most results are probably obtained in the case of planar dynamical systems, but three-dimensional dynamical systems are also a particular case. On one hand, the first step to generalize these results may be to consider three-dimensional dynamical systems and then higher dimensions. On the other hand, the increase in dimension leads to new behaviors. Finally, the parity of the dimension matters in some cases.

The aim of this special issue is to discuss the interplay of symmetries and stability in the analysis and control of three-dimensional dynamical systems and networks. Our goal is to bring out new three-dimensional mathematical models, as well as to emphasize new properties of different types of three-dimensional dynamical systems.  

Submit your paper and select the Journal “Symmetry” and the Special Issue “Three-Dimensional Dynamical Systems and Symmetry” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Dr. Cristian Lazureanu
Dr. Dana Constantinescu
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. Symmetry 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 2400 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

  • mathematical models
  • continuous-time dynamical systems
  • discrete-time dynamical systems
  • time-delay dynamical systems
  • piecewise dynamical systems
  • fractional-order dynamical systems
  • symmetries
  • integrability
  • controllability
  • stability
  • special orbits (periodic orbits and limit cycles, homoclinic and heteroclinic orbits)
  • bifurcations
  • chaotic behavior
  • numerical methods
  • numerical simulations

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Published Papers (5 papers)

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Research

21 pages, 4760 KiB  
Article
Asymmetry and Symmetry in New Three-Dimensional Chaotic Map with Commensurate and Incommensurate Fractional Orders
by Hussein Al-Taani, Ma’mon Abu Hammad, Mohammad Abudayah, Louiza Diabi and Adel Ouannas
Symmetry 2024, 16(11), 1447; https://doi.org/10.3390/sym16111447 - 31 Oct 2024
Viewed by 440
Abstract
According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and [...] Read more.
According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and incommensurate orders, revealing their impact on dynamics. Through the maximum Lyapunov exponent (LEmax), phase portraits, and bifurcation charts. In addition, we assess the complexity and confirm the chaotic features in the map using the approximation entropy ApEn and C0 complexity. Studies show that the commensurate and incommensurate derivative values influence the fractional chaotic map-based three functions, which exhibit a variety of dynamical behaviors, such as hidden attractors, asymmetry, and symmetry. Moreover, the new system’s stabilizing employing a 3D nonlinear controller is introduced. Finally, our study validates the research results using the simulation MATLAB R2024a. Full article
(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)
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23 pages, 951 KiB  
Article
Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion
by Yury Shestopalov, Azizaga Shakhverdiev and Sergey V. Arefiev
Symmetry 2024, 16(1), 14; https://doi.org/10.3390/sym16010014 - 21 Dec 2023
Cited by 1 | Viewed by 1315
Abstract
The behavior and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type-twisted fold is discovered on the basis [...] Read more.
The behavior and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type-twisted fold is discovered on the basis and within the frames of the elements of the developed DS qualitative theory. The discriminant criterion applied originally to two-phase quadratic polynomial DSs is extended to three-phase DSs investigated in terms of their coefficient matrices. Specific classes of D- and S-vectors are introduced and a complete description of the symmetry relations inherent to the DS coefficient matrices is performed using the discriminant criterion. Full article
(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)
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15 pages, 323 KiB  
Article
Short Remark on (p1,p2,p3)-Complex Numbers
by Wolf-Dieter Richter
Symmetry 2024, 16(1), 9; https://doi.org/10.3390/sym16010009 - 20 Dec 2023
Viewed by 1143
Abstract
Movements on surfaces of centered Euclidean spheres and changes between those with different radii mean complex multiplication in R3. Here, the Euclidean norm, which generates the spheres, is replaced with an inhomogeneous functional and a product is introduced in a geometric [...] Read more.
Movements on surfaces of centered Euclidean spheres and changes between those with different radii mean complex multiplication in R3. Here, the Euclidean norm, which generates the spheres, is replaced with an inhomogeneous functional and a product is introduced in a geometric analogy. Because a change in the radius now leads to a change in the shape of the sphere, a three-dimensional dynamic complex structure is created. Statements about invariant probability densities, generalized uniform distributions on generalized spheres, geometric measure representations, and dynamic ball numbers associated with this structure are also presented. Full article
(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)
13 pages, 1514 KiB  
Article
On the Bifurcations of a 3D Symmetric Dynamical System
by Dana Constantinescu
Symmetry 2023, 15(4), 923; https://doi.org/10.3390/sym15040923 - 15 Apr 2023
Cited by 2 | Viewed by 1383
Abstract
The paper studies the bifurcations that occur in the T-system, a 3D dynamical system symmetric in respect to the Oz axis. Results concerning some local bifurcations (pitchfork and Hopf bifurcation) are presented and our attention is focused on a special bifurcation, when the [...] Read more.
The paper studies the bifurcations that occur in the T-system, a 3D dynamical system symmetric in respect to the Oz axis. Results concerning some local bifurcations (pitchfork and Hopf bifurcation) are presented and our attention is focused on a special bifurcation, when the system has infinitely many equilibrium points. It is shown that, at the bifurcation limit, the phase space is foliated by infinitely many invariant surfaces, each of them containing two equilibrium points (an attractor and a saddle). For values of the bifurcation parameter close to the bifurcation limit, the study of the system’s dynamics is done according to the singular perturbation theory. The dynamics is characterized by mixed mode oscillations (also called fast-slow oscillations or oscillations-relaxations) and a finite number of equilibrium points. The specific features of the bifurcation are highlighted and explained. The influence of the pitchfork and Hopf bifurcations on the fast-slow dynamics is also pointed out. Full article
(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)
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21 pages, 698 KiB  
Article
Approximate Closed-Form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method
by Remus-Daniel Ene, Nicolina Pop and Marioara Lapadat
Symmetry 2022, 14(10), 2185; https://doi.org/10.3390/sym14102185 - 18 Oct 2022
Cited by 1 | Viewed by 1418
Abstract
Based on some geometrical properties (symmetries and global analytic first integrals) of the Rabinovich system the closed-form solutions of the equations have been established. The chaotic behaviors are excepted. Moreover, the Rabinovich system is reduced to a nonlinear differential equation depending on an [...] Read more.
Based on some geometrical properties (symmetries and global analytic first integrals) of the Rabinovich system the closed-form solutions of the equations have been established. The chaotic behaviors are excepted. Moreover, the Rabinovich system is reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions are built using the Optimal Auxiliary Functions Method (OAFM). The advantage of this method is to obtain accurate solutions for special cases, with only an analytic first integral. An important output is the existence of complex eigenvalues, depending on the initial conditions and physical parameters of the system. This approach was not still analytically emphasized from our knowledge. A good agreement between the analytical and corresponding numerical results has been performed. The accuracy of the obtained results emphasizes that this procedure could be successfully applied to more dynamic systems with these geometrical properties. Full article
(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: On regular and singular bifurcations in a 3D system with symmetry
Authors: Dana Constantinescu
Affiliation: Department of Applied Mathematics, University of Craiova, Craiova 200585, Romania

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