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Symmetry
Special Issues
Nonlinear Oscillations and Boundary Value Problems
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Nonlinear Oscillations and Boundary Value Problems

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

Deadline for manuscript submissions: closed (15 November 2020) | Viewed by 17473

Special Issue Editors

Prof. Miklós Rontó

E-Mail Website
Guest Editor
University of Miskolc, Hungary
Interests: ordinary differential equations; non-linear boundary value problems; numerical-analytic methods
Prof. Dr. András Rontó

E-Mail Website
Guest Editor
Institute of Mathematics, Czech Academy of Sciences, 61662 Brno, Czech Republic
Interests: functional differential equations; boundary value problems; numerical-analytic methods; theory of positive operators
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,
    The investigation of oscillatory phenomena is an important part of the theory of differential equations. It is well-known that oscillations occur in a natural way virtually in every area of applied science including, e.g., mechanics, electrical and radio engineering, or vibrotechnics. One can mention, for instance, the beating of the human heart in medicine, business cycles in economics, predator-prey cycles in population dynamics, vibrating strings in musical instruments, periodic firing of nerve cells in the brain. Theoretical aspects of the classical theory of oscillations include the investigation of harmonic, periodic, and almost periodic solutions of various types of ordinary differential equations and systems. Among important related tasks one should outline obtaining  sufficient conditions for the existence of such solutions, description of their asymptotic behaviour, study of oscillatory properties and mutual disposition of zeros, detection of solutions possessing particular symmetry properties, development of efficient methods for the construction of solutions. The topics mentioned have also a strong relation to the theory of non-linear boundary value problems.
    This issue is devoted to non-linear oscillations in a broad sense and will cover the related topics for non-linear systems of differential equations, equations with retarded argument and more general functional differential equations.
    We hope that researchers working in differential equations and related topics will find in this special issue new ideas and techniques that will stimulate further progress in the field.

Prof. Miklós Rontó
Dr. András Rontó
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

  • oscillatory phenomena
  • oscillatory solutions
  • sign-constant solutions
  • periodic solutions
  • non-linear boundary value problems
  • symmetry
  • numerical-analytic methods
  • functional-differential equations
  • successive approximations

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Published Papers (7 papers)

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Research

19 pages, 1004 KiB  
Article
On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type
by András Rontó and Natálie Rontóová
Symmetry 2020, 12(10), 1740; https://doi.org/10.3390/sym12101740 - 21 Oct 2020
Cited by 1 | Viewed by 1773
Abstract
We show the use of parametrization techniques and successive approximations for the effective construction of solutions of linear boundary value problems for differential systems with multiple argument deviations. The approach is illustrated with a numerical example. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
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20 pages, 887 KiB  
Article
Oscillation Properties of Singular Quantum Trees
by Monika Homa and Rostyslav Hryniv
Symmetry 2020, 12(8), 1266; https://doi.org/10.3390/sym12081266 - 1 Aug 2020
Viewed by 1999
Abstract
We discuss the possibility of generalizing the Sturm comparison and oscillation theorems to the case of singular quantum trees, that is, to Sturm-Liouville differential expressions with singular coefficients acting on metric trees and subject to some boundary and interface conditions. As there may [...] Read more.
We discuss the possibility of generalizing the Sturm comparison and oscillation theorems to the case of singular quantum trees, that is, to Sturm-Liouville differential expressions with singular coefficients acting on metric trees and subject to some boundary and interface conditions. As there may exist non-trivial solutions of differential equations on metric trees that vanish identically on some edges, the classical Sturm theory cannot hold globally for quantum trees. However, we show that the comparison theorem holds under minimal assumptions and that the oscillation theorem holds generically, that is, for operators with simple spectra. We also introduce a special Prüfer angle, establish some properties of solutions in the non-generic case, and then extend the oscillation results to simple eigenvalues. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
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9 pages, 240 KiB  
Article
On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure
by Yarema Prykarpatskyy
Symmetry 2020, 12(5), 697; https://doi.org/10.3390/sym12050697 - 1 May 2020
Cited by 1 | Viewed by 1527
Abstract
A class of spatially one-dimensional completely integrable Chaplygin hydrodynamic systems was studied within framework of Lie-algebraic approach. The Chaplygin hydrodynamic systems were considered as differential systems on the torus. It has been shown that the geometric structure of the systems under analysis has [...] Read more.
A class of spatially one-dimensional completely integrable Chaplygin hydrodynamic systems was studied within framework of Lie-algebraic approach. The Chaplygin hydrodynamic systems were considered as differential systems on the torus. It has been shown that the geometric structure of the systems under analysis has strong relationship with diffeomorphism group orbits on them. It has allowed to find a new infinite hierarchy of integrable Chaplygin like hydrodynamic systems. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
13 pages, 343 KiB  
Article
On a Class of Functional Differential Equations with Symmetries
by Nataliya Dilna, Michal Fečkan and András Rontó
Symmetry 2019, 11(12), 1456; https://doi.org/10.3390/sym11121456 - 27 Nov 2019
Cited by 4 | Viewed by 2531
Abstract
It is shown that a class of symmetric solutions of scalar non-linear functional differential equations can be investigated by using the theory of boundary value problems. We reduce the question to a two-point boundary value problem on a bounded interval and present several [...] Read more.
It is shown that a class of symmetric solutions of scalar non-linear functional differential equations can be investigated by using the theory of boundary value problems. We reduce the question to a two-point boundary value problem on a bounded interval and present several conditions ensuring the existence of a unique symmetric solution. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
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10 pages, 259 KiB  
Article
Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders
by Josef Rebenda
Symmetry 2019, 11(11), 1390; https://doi.org/10.3390/sym11111390 - 9 Nov 2019
Cited by 4 | Viewed by 2188
Abstract
The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a [...] Read more.
The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem gives a reliable and expected outcome including the phenomenon of symmetry in choice of orders of the differential transformation of the multi-order system. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
14 pages, 2715 KiB  
Article
Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain
by Xiuying Guo, Gang Zhang and Ruilan Tian
Symmetry 2019, 11(7), 886; https://doi.org/10.3390/sym11070886 - 6 Jul 2019
Cited by 12 | Viewed by 4121
Abstract
In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for [...] Read more.
In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
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14 pages, 2096 KiB  
Article
Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method
by Zhiwei Zhang, Yingjie Wang, Wei Wang and Ruilan Tian
Symmetry 2019, 11(5), 676; https://doi.org/10.3390/sym11050676 - 16 May 2019
Cited by 8 | Viewed by 2626
Abstract
In this article, we present a new accurate iterative and asymptotic method to construct analytical periodic solutions for a strongly nonlinear system, even if it is not Z2-symmetric. This method is applicable not only to a conservative system but also to [...] Read more.
In this article, we present a new accurate iterative and asymptotic method to construct analytical periodic solutions for a strongly nonlinear system, even if it is not Z2-symmetric. This method is applicable not only to a conservative system but also to a non-conservative system with a limit cycle response. Distinct from the general harmonic balance method, it depends on balancing a few trigonometric terms (at most five terms) in the energy equation of the nonlinear system. According to this iterative approach, the dynamic frequency is a trigonometric function that varies with time t, which represents the influence of derivatives of the higher harmonic terms in a compact form and leads to a significant reduction of calculation workload. Two examples were solved and numerical solutions are presented to illustrate the effectiveness and convenience of the method. Based on the present method, we also outline a modified energy balance method to further simplify the procedure of higher order computation. Finally, a nonlinear strength index is introduced to automatically identify the strength of nonlinearity and classify the suitable strategies. Full article
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
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