Special Issue "Asymptotic Methods in the Mechanics and Nonlinear Dynamics"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 31 August 2020.

Special Issue Editor

Prof. Igor Andrianov
E-Mail Website
Guest Editor
Karl-Marx-Allee 4, D-50769, Koeln, Germany.
Interests: Asymptotology; Mechanics of Solids; Nonlinear Dynamics

Special Issue Information

Dear Colleagues,

This Special Issue of Symmetry is dedicated to asymptotic modelling. Galileo’s concept of idealization is the cornerstone of contemporary science. Idealization is based on increasing the symmetry of the original system, and its tool in applied mathematics is asymptotic analysis. It is no exaggeration to say that the basic models of applied mathematics, physics, and mechanics are asymptotic. There are many methods for constructing asymptotic models, and their development, generalization, and application are of fundamental importance both for theorists and for engineers. We hope this Special Issue will help theorists to find new tasks and areas of application for their knowledge, and engineers to find new methods for their practice.

This Special Issue invites research and review papers on various fields of theoretical physics and applied mathematics, including classical and quantum mechanics, mechanics of fluids and solids, and asymptotology.

Prof. Igor Andrianov
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 papers will be 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 1400 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

  • Asymptotology
  • asymptotic modelling
  • regular and singular perturbation problems
  • Padé approximants and others summation and interpolation procedures
  • homogenization
  • kontinualization

Published Papers (4 papers)

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Research

Open AccessArticle
Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation
Symmetry 2020, 12(1), 131; https://doi.org/10.3390/sym12010131 - 09 Jan 2020
Abstract
We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi [...] Read more.
We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
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Open AccessArticle
The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior
Symmetry 2019, 11(12), 1446; https://doi.org/10.3390/sym11121446 - 25 Nov 2019
Abstract
In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O Δ E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system [...] Read more.
In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O Δ E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system with very complicated (chaotic) behavior. In our paper we present examples of deterministic discretization and chaotic continualization. Continualization procedure is based on Padé approximants. To correctly characterize the dynamics of obtained ODE we measured such characteristic parameters of chaotic dynamical systems as the Lyapunov exponents and the Lyapunov dimensions. Discretization and continualization lead to a change in the symmetry of the mathematical model (i.e., group properties of the original ODE and O Δ E). This aspect of the problem is the aim of further research. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
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Open AccessArticle
Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine
Symmetry 2019, 11(12), 1444; https://doi.org/10.3390/sym11121444 - 23 Nov 2019
Abstract
This paper addresses a second-order sliding mode control method for the formation problem of multirobot systems. The formation patterns are usually symmetrical. This sliding mode control is based on the super-twisting law. In many real-world applications, the robots suffer from a great diversity [...] Read more.
This paper addresses a second-order sliding mode control method for the formation problem of multirobot systems. The formation patterns are usually symmetrical. This sliding mode control is based on the super-twisting law. In many real-world applications, the robots suffer from a great diversity of uncertainties and disturbances that greatly challenge super-twisting sliding mode formation maneuvers. In particular, such a challenge has adverse effects on the formation performance when the uncertainties and disturbances have an unknown bound. This paper focuses on this issue and utilizes the technique of an extreme learning machine to meet this challenge. Within the leader–follower framework, this paper investigates the integration of the super-twisting sliding mode control method and the extreme learning machine. The output weights of this extreme learning machine are adaptively adjusted so that this integrated formation design has guaranteed closed-loop stability in the sense of Lyaponov. In the end, some simulations are implemented via a multirobot platform, illustrating the superiority and effectiveness of the integrated formation design in spite of uncertainties and disturbances. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
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Open AccessArticle
Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations
Symmetry 2019, 11(11), 1420; https://doi.org/10.3390/sym11111420 - 16 Nov 2019
Abstract
Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in [...] Read more.
Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in terms of elementary functions. As a result, closed form analytical solutions for the non-autonomous strongly nonlinear case are obtained. Conditions of existence for such solutions are represented as a family of period-amplitude curves. The family is represented by different couples of solutions associated with different numbers of vibration half cycles between any two consecutive pulses. Poincaré sections showed that the oscillator can respond quite chaotically when shifting from the period-amplitude curves. Full article
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
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