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Symmetry
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Qualitative Theory and Symmetries of Ordinary Differential Equations
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Qualitative Theory and Symmetries of Ordinary Differential Equations

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

Deadline for manuscript submissions: closed (31 January 2022) | Viewed by 11319

Special Issue Editor

Prof. Dr. Jaume Giné

E-Mail Website
Guest Editor
Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69. 25001 Lleida, Catalonia, Spain
Interests: qualitative theory; ordinary differential equations; integrability; bifurcation and dynamical systems

Special Issue Information

Dear Colleagues,

Qualitative analysis has proved to be an important and useful tool for investigating the properties of solutions of differential equations, because it is able to analyze differential equations without solving them analytically and numerically. Since the qualitative analysis of differential equations is related to both pure and applied mathematics, its applications to various fields such as science, engineering, and ecology have been extensively developed.

The objective of this Special Issue is to report on the latest achievements in the qualitative theory of ordinary differential equations and their relation to the symmetries of differential equations. It will reflect both the state-of-the-art theoretical research and important recent advances in applications. It is important to develop new theories and methods, as well as to modify and refine the well-known techniques for the analysis of new classes of problems using the symmetries of differential equations. We are mainly interested in ordinary differential equations, autonomous or non-autonomous, smooth or non-smooth. We hope to gather together established and young scientists actively working on the subject.

This Special Issue will collect high-quality contributions from leading experts and researchers actively working in the field. The topics of interest include, but are not limited to the singularities and local behavior of solutions, the stability properties and asymptotic behavior of solutions, existence, the bifurcations and stability of periodic solutions, the existence and properties of almost-periodic solutions, nonlinear ordinary differential operators, and the symmetries and integrability of ordinary differential equations.

Prof. Dr. Jaume Giné
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

  • qualitative theory
  • stability
  • bifurcation
  • singular points
  • periodic solutions
  • symmetries
  • integrability

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Published Papers (6 papers)

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Research

17 pages, 425 KiB  
Article
On the Dynamics of Higgins–Selkov, Selkov and Brusellator Oscillators
by Jaume Giné
Symmetry 2022, 14(3), 438; https://doi.org/10.3390/sym14030438 - 23 Feb 2022
Cited by 1 | Viewed by 1543
Abstract
A complete algebraic characterization of the first integrals of the Higgins–Selkov, Selkov and Brusellator oscillators is given here. The existence of symmetries sometimes forces the existence of such first integrals. The nonexistence of centers for such oscillators is also proved. In order to [...] Read more.
A complete algebraic characterization of the first integrals of the Higgins–Selkov, Selkov and Brusellator oscillators is given here. The existence of symmetries sometimes forces the existence of such first integrals. The nonexistence of centers for such oscillators is also proved. In order to determine the Puiseux integrability of such systems, the multiple Puiseux solutions are also studied. Full article
(This article belongs to the Special Issue Qualitative Theory and Symmetries of Ordinary Differential Equations)
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21 pages, 420 KiB  
Article
Integrability and Limit Cycles via First Integrals
by Jaume Llibre
Symmetry 2021, 13(9), 1736; https://doi.org/10.3390/sym13091736 - 18 Sep 2021
Cited by 8 | Viewed by 2120
Abstract
In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely [...] Read more.
In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems. Full article
(This article belongs to the Special Issue Qualitative Theory and Symmetries of Ordinary Differential Equations)
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25 pages, 426 KiB  
Article
Linearizability of 2:−3 Resonant Systems with Quadratic Nonlinearities
by Maja Žulj, Brigita Ferčec and Matej Mencinger
Symmetry 2021, 13(8), 1510; https://doi.org/10.3390/sym13081510 - 17 Aug 2021
Cited by 2 | Viewed by 1378
Abstract
In this paper, the linearizability of a 2:−3 resonant system with quadratic nonlinearities is studied. We provide a list of the conditions for this family of systems having a linearizable center. The conditions for linearizablity are obtained by computing the ideal generated by [...] Read more.
In this paper, the linearizability of a 2:−3 resonant system with quadratic nonlinearities is studied. We provide a list of the conditions for this family of systems having a linearizable center. The conditions for linearizablity are obtained by computing the ideal generated by the linearizability quantities and its decomposition into associate primes. To successfully perform the calculations, we use an approach based on modular computations. The sufficiency of the obtained conditions is proven by several methods, mainly by the method of Darboux linearization. Full article
(This article belongs to the Special Issue Qualitative Theory and Symmetries of Ordinary Differential Equations)
28 pages, 399 KiB  
Article
Orbital Hypernormal Forms
by Antonio Algaba, Estanislao Gamero and Cristóbal García
Symmetry 2021, 13(8), 1500; https://doi.org/10.3390/sym13081500 - 16 Aug 2021
Cited by 1 | Viewed by 1666
Abstract
In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the [...] Read more.
In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularity. Full article
(This article belongs to the Special Issue Qualitative Theory and Symmetries of Ordinary Differential Equations)
10 pages, 294 KiB  
Article
Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles
by Jaume Llibre and Claudia Valls
Symmetry 2021, 13(7), 1128; https://doi.org/10.3390/sym13071128 - 24 Jun 2021
Cited by 14 | Viewed by 2029
Abstract
We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or [...] Read more.
We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1. Full article
(This article belongs to the Special Issue Qualitative Theory and Symmetries of Ordinary Differential Equations)
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10 pages, 266 KiB  
Article
New Aspects for Oscillation of Differential Systems with Mixed Delays and Impulses
by Shyam Sundar Santra, Khaled Mohamed Khedher and Shao-Wen Yao
Symmetry 2021, 13(5), 780; https://doi.org/10.3390/sym13050780 - 1 May 2021
Cited by 9 | Viewed by 1398
Abstract
Oscillation and symmetry play an important role in many applications such as engineering, physics, medicine, and vibration in flight. In this work, we obtain sufficient and necessary conditions for the oscillation of the solutions to a second-order differential equation with impulses and mixed [...] Read more.
Oscillation and symmetry play an important role in many applications such as engineering, physics, medicine, and vibration in flight. In this work, we obtain sufficient and necessary conditions for the oscillation of the solutions to a second-order differential equation with impulses and mixed delays when the neutral coefficient lies within [0,1). Furthermore, an examination of the validity of the proposed criteria has been demonstrated via particular examples. Full article
(This article belongs to the Special Issue Qualitative Theory and Symmetries of Ordinary Differential Equations)
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