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Symmetry
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Symmetry in Discrete Dynamical Systems and Ordinary Differential Equations
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Symmetry in Discrete Dynamical Systems and 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 August 2020) | Viewed by 13741

Special Issue Editors

Prof. Dr. Valery G. Romanovski

E-Mail Website
Guest Editor
Faculty of Electrical Engineering, University of Maribor and the Center for Applied Mathematics and Theoretical Physics, 2000 Maribor, Slovenia
Interests: qualitative theory of ordinary differential equations and the theory of bifurcations of plane dynamical systems; integrability of dynamical systems; symbolic computation; computational algebra and algebraic geometry
Prof. Dr. Xingwu Chen

E-Mail Website
Co-Guest Editor
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Interests: qualitative theory; bifurcation theory; ODEs; dynamical systems; nonlinear dynamics

Special Issue Information

Dear Colleagues,

Nowadays, numerous real world phenomena and processes are mathematically modelled by discrete dynamical systems and ordinary differential equations. Many of such models exhibit different kinds of symmetry. It can be symmetry of discrete or differential equations with respect to some groups of transformation, symmetry of phase portraits, symmetry of certain solutions, time-reversible symmetry etc. Sometimes symmetry is hidden and can be seen only after some nonlinear transformations. Some kinds of symmetry in dynamical systems are often related to specific properties of the systems, like integrability, linearizability and periodicity of solutions. Existence of symmetry can be helpful in the control theory since it can allow to control real world models described by ordinary differential equations or discrete dynamical equations. Sometimes symmetry can be used in order to reduce a system of differential equations to an equivalent system of a simple form. It is also important on studies on Hilbert’s 16th problem, since it allows to construct polynomial systems of ODEs with many limit cycles.

You are welcome to submit your paper related to the topics mentioned above for the Special Issue.

Prof. Dr. Valery G. Romanovski
Prof. Dr. Xingwu Chen
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

  • symmetry with respect to linear groups
  • ODEs
  • time-reversible symmetry
  • solutions and phase portraits
  • integrability
  • linearizability
  • periodic solutions
  • symbolic and numeric methods for systems with symmetry

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Published Papers (5 papers)

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Research

12 pages, 1184 KiB  
Article
Limit Cycle Bifurcations Near a Cuspidal Loop
by Pan Liu and Maoan Han
Symmetry 2020, 12(9), 1425; https://doi.org/10.3390/sym12091425 - 27 Aug 2020
Cited by 2 | Viewed by 1793
Abstract
In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near [...] Read more.
In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center. Full article
(This article belongs to the Special Issue Symmetry in Discrete Dynamical Systems and Ordinary Differential Equations)
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19 pages, 804 KiB  
Article
On Some Symmetries of Quadratic Systems
by Maoan Han, Tatjana Petek and Valery G. Romanovski
Symmetry 2020, 12(8), 1300; https://doi.org/10.3390/sym12081300 - 4 Aug 2020
Cited by 1 | Viewed by 2082
Abstract
We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, [...] Read more.
We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented. Full article
(This article belongs to the Special Issue Symmetry in Discrete Dynamical Systems and Ordinary Differential Equations)
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21 pages, 1213 KiB  
Article
Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces
by Jan Awrejcewicz, Nataliya Losyeva and Volodymyr Puzyrov
Symmetry 2020, 12(8), 1210; https://doi.org/10.3390/sym12081210 - 23 Jul 2020
Cited by 13 | Viewed by 2116
Abstract
We consider a linear dynamical system under the action of potential and circulatory forces. The matrix of potential forces is positive definite, and the main question is when the circulatory forces induce instability to the system. Different approaches to studying the problem are [...] Read more.
We consider a linear dynamical system under the action of potential and circulatory forces. The matrix of potential forces is positive definite, and the main question is when the circulatory forces induce instability to the system. Different approaches to studying the problem are discussed and illustrated by examples. The case of multiple eigenvalues also is considered, and sufficient conditions of instability are obtained. Some issues of the dynamics of a nonlinear system with an unstable linear approximation are discussed. The behavior of trajectories in the case of unstable equilibrium is investigated, and an example of the chaotic behavior versus the case of bounded solutions is presented and discussed. Full article
(This article belongs to the Special Issue Symmetry in Discrete Dynamical Systems and Ordinary Differential Equations)
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12 pages, 715 KiB  
Article
Symmetries in Phase Portrait
by Yakov Krasnov and Umbetkul K. Koylyshov
Symmetry 2020, 12(7), 1123; https://doi.org/10.3390/sym12071123 - 6 Jul 2020
Cited by 3 | Viewed by 2585
Abstract
We construct polynomial dynamical systems x ′ = P ( x ) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis. [...] Read more.
We construct polynomial dynamical systems x ′ = P ( x ) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis. Full article
(This article belongs to the Special Issue Symmetry in Discrete Dynamical Systems and Ordinary Differential Equations)
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22 pages, 12153 KiB  
Article
Criticality or Supersymmetry Breaking?
by Igor V. Ovchinnikov, Wenyuan Li, Yuquan Sun, Andrew E. Hudson, Karlheinz Meier, Robert N. Schwartz and Kang L. Wang
Symmetry 2020, 12(5), 805; https://doi.org/10.3390/sym12050805 - 12 May 2020
Cited by 6 | Viewed by 4392
Abstract
In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1/f-type power spectra and/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found [...] Read more.
In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1/f-type power spectra and/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found approximation-free supersymmetric theory of stochastics (STS) identifies this phase as the noise-induced chaos (N-phase), i.e., the phase where the topological supersymmetry pertaining to all stochastic dynamical systems is broken spontaneously by the condensation of the noise-induced (anti)instantons. Here, we support this picture in the context of neurodynamics. We study a 1D chain of neuron-like elements and find that the dynamics in the N-phase is indeed featured by positive stochastic Lyapunov exponents and dominated by (anti)instantonic processes of (creation) annihilation of kinks and antikinks, which can be viewed as predecessors of boundaries of neuroavalanches. We also construct the phase diagram of emulated stochastic neurodynamics on Spikey neuromorphic hardware and demonstrate that the width of the N-phase vanishes in the deterministic limit in accordance with STS. As a first result of the application of STS to neurodynamics comes the conclusion that a conscious brain can reside only in the N-phase. Full article
(This article belongs to the Special Issue Symmetry in Discrete Dynamical Systems and Ordinary Differential Equations)
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