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Symmetry
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Advances in Nonlinear Systems and Symmetry/Asymmetry
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Advances in Nonlinear Systems and Symmetry/Asymmetry

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

Deadline for manuscript submissions: 31 July 2025 | Viewed by 2104

Special Issue Editors

Dr. Nannan Zhao

E-Mail Website
Guest Editor
School of Science, Chang’an University, Xi’an 710064, China
Interests: coupled oscillators; nonlinear dynamics; complex network; theoretical ecology; multi-agent system; pattern formation
Prof. Dr. Zhongkui Sun

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, China
Interests: nonlinear dynamics; vibration control; epidemics; complex systems; humanoid robots

Special Issue Information

Dear Colleagues,

Nonlinear systems, which are prevalent across various natural and man-made systems, exhibit complex behaviors such as bifurcations, chaos, and stability, making their study crucial for numerous applications. A rich variety of collective dynamical states and patterns have been unveiled when nonlinear dynamical systems meet with symmetry and/or asymmetry. In general, symmetry in nonlinear systems, commonly found in many natural and engineered settings, simplifies their analysis and often leads to more predictable behaviors, which is critical in understanding pattern formation and system evolution for fluid dynamics, population dynamics, and epidemics. On the one hand, asymmetric nonlinear systems will display a wider range of behaviors and are often more challenging to analyze. These systems highlight the impact of asymmetry on performance and stability, offering insights into real-world phenomena.

The Special Issue encourages researchers to contribute their original and high-quality works that advance the understanding of nonlinear systems with symmetry/asymmetry. Potential topics include, but are not limited to, the following: synchronization/chimera state of coupled oscillators; pattern formation in physical, chemical, and biological systems; and other interdisciplinary applications such as dynamics and stability in quantum systems, mechanical systems, and electrical circuits.

Dr. Nannan Zhao
Prof. Dr. Zhongkui Sun
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

  • nonlinear systems
  • coupled oscillators
  • complex networks
  • complex systems
  • synchronization
  • pattern formation
  • phase transitions

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (2 papers)

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Research

20 pages, 304 KiB  
Article
Derivation of Tensor Algebra as a Fundamental Operation—The Fermi Derivative in a General Metric Affine Space
by Michael Tsamparlis
Symmetry 2025, 17(1), 81; https://doi.org/10.3390/sym17010081 - 7 Jan 2025
Viewed by 700
Abstract
The aim of this work is to demonstrate that all linear derivatives of the tensor algebra over a smooth manifold M can be viewed as specific cases of a broader concept—the operation of derivation. This approach reveals the universal role of differentiation, which [...] Read more.
The aim of this work is to demonstrate that all linear derivatives of the tensor algebra over a smooth manifold M can be viewed as specific cases of a broader concept—the operation of derivation. This approach reveals the universal role of differentiation, which simplifies and generalizes the study of tensor derivatives, making it a powerful tool in Differential Geometry and related fields. To perform this, the generic derivative is introduced, which is defined in terms of the quantities Qk(i)(X). Subsequently, the transformation law of these quantities is determined by the requirement that the generic derivative of a tensor is a tensor. The quantities Qk(i)(X) and their transformation law define a specific geometric object on M, and consequently, a geometric structure on M. Using the generic derivative, one defines the tensor fields of torsion and curvature and computes them for all linear derivatives in terms of the quantities Qk(i)(X). The general model is applied to the cases of Lie derivative, covariant derivative, and Fermi derivative. It is shown that the Lie derivative has non-zero torsion and zero curvature due to the Jacobi identity. For the covariant derivative, the standard results follow without any further calculations. Concerning the Fermi derivative, this is defined in a new way, i.e., as a higher-order derivative defined in terms of two derivatives: a given derivative and the Lie derivative. Being linear derivative, it has torsion and curvature tensor. These fields are computed in a general affine space from the corresponding general expressions of the generic derivative. Applications of the above considerations are discussed in a number of cases. Concerning the Lie derivative, it is been shown that the Poisson bracket is in fact a Lie derivative. Concerning the Fermi derivative, two applications are considered: (a) the explicit computation of the Fermi derivative in a general affine space and (b) the consideration of Freedman–Robertson–Walker spacetime endowed with a scalar torsion field, which satisfies the Cosmological Principle and the computation of Fermi derivative of the spatial directions defining a spatial frame along the cosmological fluid of comoving observers. It is found that torsion, even in this highly symmetric case, induces a kinematic rotation of the space axes, questioning the interpretation of torsion as a spin. Finally it is shown that the Lie derivative of the dynamical equations of an autonomous conservative dynamical system is equivalent to the standard Lie symmetry method. Full article
(This article belongs to the Special Issue Advances in Nonlinear Systems and Symmetry/Asymmetry)
19 pages, 17654 KiB  
Article
New Processing Technique of Jacobian Elliptic Equation and Its Application to the (3+1)-Dimensional Modified Korteweg de Vries–Zakharov–Kuznetsov Equation
by Guojiang Wu, Yong Guo and Yanlin Yu
Symmetry 2024, 16(10), 1285; https://doi.org/10.3390/sym16101285 - 30 Sep 2024
Viewed by 1056
Abstract
This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, [...] Read more.
This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, a large number of new types of exact periodic wave solutions were obtained, many of which were rarely found in previous research. Numerical simulations have demonstrated the evolution of various periodic waves in (3+1)-dimensional mKDV-ZK. The solutions and wave phenomena obtained in this article will help expand our understanding of the equation. Full article
(This article belongs to the Special Issue Advances in Nonlinear Systems and Symmetry/Asymmetry)
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