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Symmetry
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Selected Papers on Nonlinear Dynamics
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Selected Papers on Nonlinear Dynamics

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

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 6254

Special Issue Editor

Prof. Dr. Branko Dragovich

E-Mail Website
Guest Editor
1. Institute of Physics, University of Belgrade, Belgrade, Serbia
2. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia
Interests: modified gravity; cosmology; p-adic analysis; p-adic mathematical physics; p-adic string theory; genetic code and bioinformation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The 3rd Conference on Nonlinearity is an international conference that will be held on Sep 4—8, 2023. This is the third conference in a series of conferences on nonlinearity that we plan to continue in the future within a period of two years. The 1st Conference on Nonlinearity was held on Oct 11-12, 2019 in Belgrade, Serbia, and the the 2nd Conference on Nonlinearity was held on Oct 18-22, 2021. The main organizer is the Serbian Academy of Nonlinear Sciences (SANS). There are also several coorganizers from Serbia.

We cordially invite researchers working in these fields to contribute original research papers or review articles to this Special Issue.

Prof. Dr. Branko Dragovich
Guest Editor

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 dynamics
  • nonlinear optics
  • nonlinear differential equations
  • nonlinear metaphotonics
  • solitons
  • nonlinear Schroedinger equation
  • nonlinear analysis
  • self-organization

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Published Papers (5 papers)

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Research

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16 pages, 2669 KiB  
Article
Stable Patterns in the Lugiato–Lefever Equation with a Confined Vortex Pump
by Shatrughna Kumar, Wesley B. Cardoso and Boris A. Malomed
Symmetry 2024, 16(4), 470; https://doi.org/10.3390/sym16040470 - 12 Apr 2024
Cited by 1 | Viewed by 831
Abstract
We introduce a model of a passive optical cavity based on a novel variety of the two-dimensional Lugiato–Lefever equation, with a localized pump carrying intrinsic vorticity S, and the cubic or cubic–quintic nonlinearity. Up to S=5, stable confined vortex [...] Read more.
We introduce a model of a passive optical cavity based on a novel variety of the two-dimensional Lugiato–Lefever equation, with a localized pump carrying intrinsic vorticity S, and the cubic or cubic–quintic nonlinearity. Up to S=5, stable confined vortex ring states (vortex pixels) are produced by means of a variational approximation and in a numerical form. Surprisingly, vast stability areas of the vortex states are found, for both the self-focusing and defocusing signs of the nonlinearity, in the plane of the pump and loss parameters. When the vortex rings are unstable, they are destroyed by azimuthal perturbations, which break the axial symmetry. The results suggest new possibilities for mode manipulations in passive nonlinear photonic media by means of appropriately designed pump beams. Full article
(This article belongs to the Special Issue Selected Papers on Nonlinear Dynamics)
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16 pages, 587 KiB  
Article
Constraints on Graviton Mass from Schwarzschild Precession in the Orbits of S-Stars around the Galactic Center
by Predrag Jovanović, Vesna Borka Jovanović, Duško Borka and Alexander F. Zakharov
Symmetry 2024, 16(4), 397; https://doi.org/10.3390/sym16040397 - 28 Mar 2024
Cited by 2 | Viewed by 1317
Abstract
In this paper we use a modification of the Newtonian gravitational potential with a non-linear Yukawa-like correction, as it was proposed by C. Will earlier to obtain new bounds on graviton mass from the observed orbits of S-stars around the Galactic Center (GC). [...] Read more.
In this paper we use a modification of the Newtonian gravitational potential with a non-linear Yukawa-like correction, as it was proposed by C. Will earlier to obtain new bounds on graviton mass from the observed orbits of S-stars around the Galactic Center (GC). This phenomenological potential differs from the gravitational potential obtained in the weak field limit of Yukawa gravity, which we used in our previous studies. We also assumed that the orbital precession of S-stars is close to the prediction of General Relativity (GR) for Schwarzschild precession, but with a possible small discrepancy from it. This assumption is motivated by the fact that the GRAVITY Collaboration in 2020 and in 2022 detected Schwarzschild precession in the S2 star orbit around the Supermassive Black Hole (SMBH) at the GC. Using this approach, we were able to constrain parameter λ of the potential and, assuming that it represents the graviton Compton wavelength, we also found the corresponding upper bound of graviton mass. The obtained results were then compared with our previous estimates, as well as with the estimates of other authors. Full article
(This article belongs to the Special Issue Selected Papers on Nonlinear Dynamics)
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14 pages, 1755 KiB  
Article
Partial Control and Beyond: Controlling Chaotic Transients with the Safety Function
by Rubén Capeáns and Miguel A. F. Sanjuan
Symmetry 2024, 16(3), 338; https://doi.org/10.3390/sym16030338 - 11 Mar 2024
Viewed by 1280
Abstract
Chaotic dynamical systems often exhibit transient chaos, where trajectories behave chaotically for a short amount of time before escaping to an external attractor. Sustaining transient chaotic dynamics under disturbances is challenging yet desirable for many applications. The partial control approach exploits the inherent [...] Read more.
Chaotic dynamical systems often exhibit transient chaos, where trajectories behave chaotically for a short amount of time before escaping to an external attractor. Sustaining transient chaotic dynamics under disturbances is challenging yet desirable for many applications. The partial control approach exploits the inherent symmetry and geometric structure of chaotic saddles, the topological object responsible of transient chaos, to enable surprising control with only small perturbations. Here, we review the latest findings in partial control techniques with the aim to sustain chaos or accelerate escapes by exploiting these intricate invariant sets. We introduce the fundamental concept of safe sets regions where orbits persist despite noise. This paper presents recent generalizations through safety functions and escape functions that automatically find the minimum control needed. Efficient numerical algorithms are presented and several examples of application are illustrated. Rather than eliminating chaos entirely, partial control techniques provide a framework to reliably control transient chaotic dynamics with minimal interventions. This approach has promising applications across diverse fields including physics, engineering, biology, and more. Full article
(This article belongs to the Special Issue Selected Papers on Nonlinear Dynamics)
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12 pages, 262 KiB  
Article
Optimal Choice of the Auxiliary Equation for Finding Symmetric Solutions of Reaction–Diffusion Equations
by Carmen Ionescu and Radu Constantinescu
Symmetry 2024, 16(3), 335; https://doi.org/10.3390/sym16030335 - 11 Mar 2024
Cited by 1 | Viewed by 970
Abstract
This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two [...] Read more.
This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case. Full article
(This article belongs to the Special Issue Selected Papers on Nonlinear Dynamics)

Review

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27 pages, 933 KiB  
Review
Non-Local Cosmology: From Theory to Observations
by Francesco Bajardi and Salvatore Capozziello
Symmetry 2024, 16(5), 579; https://doi.org/10.3390/sym16050579 - 8 May 2024
Viewed by 1289
Abstract
We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function [...] Read more.
We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function F(R,□−1R), where R denotes the Ricci curvature scalar and the operator □−1 introduces non-locality. By selecting the functional forms using Noether Symmetries, we identify exact solutions within a cosmological framework. We can thus reduce the dynamics of these chosen models and obtain analytical solutions for the equations of motion. Therefore, we study the capability of the selected models in matching cosmological observations by evaluating the phase space and the fixed points; this allows one to further constrain the non-local model selected by symmetry considerations. Furthermore, we also investigate gravitational non-local effects on astrophysical scales. In this context, we seek symmetries within the framework of f(R,□−1R) gravity and place constraints on the free parameters. Specifically, we analyze the impact of non-locality on the orbits of the S2 star orbiting SgrA*. Full article
(This article belongs to the Special Issue Selected Papers on Nonlinear Dynamics)
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