Special Issue "Symmetry and Dynamical Systems"

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

Deadline for manuscript submissions: closed (31 March 2020).

Special Issue Editor

Prof. Viorel Nitica
Website
Guest Editor
West Chester University of Pennsylvania: West Chester, PA
Interests: dynamical systems; rigidity of smooth group actions; ergodic theory; extremal algebras; operator algebras

Special Issue Information

Dear Colleagues,

     The theory of dynamical systems is one of the cornerstones of contemporary mathematics, with connections and applications to various other major fields, such as number theory, analysis, probability theory, and statistics.

     The goal of this Special Issue is to present the various aspects of the field of dynamical systems. The two main contemporary themes are the study of generic behavior in large and thin classes of transformations and the study of rigid dynamical systems. Rigidity appears many times in the presence of a large group of symmetries of the dynamical system. A large group may be one that contains a higher rank Abelian subgroup.

Prof. Viorel Nitica
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.

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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

  • topologically transitive
  • topologically mixing
  • periodic point
  • extension
  • cocycle
  • coboundary
  • rigidity
  • recurrence
  • ergodicity
  • Lyapunov exponent
  • hyperbolicity
  • stability
  • generic behavior
  • minimality
  • infinite dymensional dynamics

Published Papers (11 papers)

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Research

Open AccessArticle
On a Semigroup Problem II
Symmetry 2020, 12(9), 1392; https://doi.org/10.3390/sym12091392 - 21 Aug 2020
Abstract
We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer [...] Read more.
We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
Open AccessArticle
Observability and Symmetries of Linear Control Systems
Symmetry 2020, 12(6), 953; https://doi.org/10.3390/sym12060953 - 04 Jun 2020
Abstract
The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a connected Lie group G. For that, [...] Read more.
The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
Open AccessArticle
Critically-Finite Dynamics on the Icosahedron
Symmetry 2020, 12(1), 177; https://doi.org/10.3390/sym12010177 - 19 Jan 2020
Abstract
A recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how [...] Read more.
A recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how to build such “dynamical polyhedra” for other icosahedral maps. First, algebra associated with the icosahedron determines a special family of maps with 60 periodic critical points. The topological behavior of each map is then worked out and results in a geometric algorithm out of which emerges a system of edges—the dynamical polyhedron—in natural correspondence to a map’s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps’ geometric behavior fall into combinatorial classes the presentation of which concludes the paper. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
Nonlinear Dynamic Modelling of Two-Point and Symmetrically Supported Pipeline Brackets with Elastic-Porous Metal Rubber Damper
Symmetry 2019, 11(12), 1479; https://doi.org/10.3390/sym11121479 - 04 Dec 2019
Abstract
This paper aims to investigate the nonlinear dynamic properties of a two-point and symmetrically supported pipeline bracket system coated with the damping element using an elastic-porous metal rubber. The dynamic model of the studied two-point and symmetric pipeline system was established based on [...] Read more.
This paper aims to investigate the nonlinear dynamic properties of a two-point and symmetrically supported pipeline bracket system coated with the damping element using an elastic-porous metal rubber. The dynamic model of the studied two-point and symmetric pipeline system was established based on impulse response matrix for accurate and reliable description on its nonlinear behaviours, e.g., energy dissipation and loss factor. The experimental verification of the developed model was performed by means of dynamic test as well as the analyses of nonlinear damping characteristics. The experimental results show a good agreement with the prediction results obtained from the proposed dynamic model. This work provides an alternative method to investigate the dynamics of pipeline vibration system equipped with a damping structure. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
An E-Sequence Approach to the 3x + 1 Problem
Symmetry 2019, 11(11), 1415; https://doi.org/10.3390/sym11111415 - 15 Nov 2019
Abstract
For any odd positive integer x, define ( x n ) n 0 and ( a n ) n 1 by setting x 0 = x ,   x n = 3 x n 1 + 1 2 a [...] Read more.
For any odd positive integer x, define ( x n ) n 0 and ( a n ) n 1 by setting x 0 = x ,   x n = 3 x n 1 + 1 2 a n such that all x n are odd. The 3 x + 1 problem asserts that there is an x n = 1 for all x. Usually, ( x n ) n 0 is called the trajectory of x. In this paper, we concentrate on ( a n ) n 1 and call it the E-sequence of x. The idea is that we generalize E-sequences to all infinite sequences ( a n ) n 1 of positive integers and consider all these generalized E-sequences. We then define ( a n ) n 1 to be Ω -convergent to x if it is the E-sequence of x and to be Ω -divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the Ω -divergence of all non-periodic E-sequences implies the periodicity of ( x n ) n 0 for all x 0 . The principal results of this paper are to prove the Ω -divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences ( a n ) n 1 with lim ¯ n b n n > log 2 3 are Ω -divergent by using Wendel’s inequality and the Matthews and Watts’ formula x n = 3 n x 0 2 b n k = 0 n 1 ( 1 + 1 3 x k ) , where b n = k = 1 n a k . These results present a possible way to prove the periodicity of trajectories of all positive integers in the 3 x + 1 problem, and we call it the E-sequence approach. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
Open AccessArticle
About the Orbit Structure of Sequences of Maps of Integers
Symmetry 2019, 11(11), 1374; https://doi.org/10.3390/sym11111374 - 06 Nov 2019
Abstract
Motivated by connections to the study of sequences of integers, we study, from a dynamical systems point of view, the orbit structure for certain sequences of maps of integers. We find sequences of maps for which all individual orbits are bounded and periodic [...] Read more.
Motivated by connections to the study of sequences of integers, we study, from a dynamical systems point of view, the orbit structure for certain sequences of maps of integers. We find sequences of maps for which all individual orbits are bounded and periodic and for which the number of periodic orbits of fixed period is finite. This allows the introduction of a formal ζ -function for the maps in these sequences, which are actually polynomials. We also find sequences of maps for which the orbit structure is more complicated, as they have both bounded and unbounded orbits, both individual and global. Most of our results are valid in a general numeration base. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
Some Chaos Notions on Dendrites
Symmetry 2019, 11(10), 1309; https://doi.org/10.3390/sym11101309 - 17 Oct 2019
Cited by 1
Abstract
Transitivity is a key element in a chaotic dynamical system. In this paper, we present some relations between transitivity, stronger and alternative notions of it on compact and dendrite spaces. The relation between Auslander and Yorke chaos and Devaney chaos on dendrites is [...] Read more.
Transitivity is a key element in a chaotic dynamical system. In this paper, we present some relations between transitivity, stronger and alternative notions of it on compact and dendrite spaces. The relation between Auslander and Yorke chaos and Devaney chaos on dendrites is also discussed. Moreover, we prove that Devaney chaos implies strong dense periodicity on dendrites while the converse is not true. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
Filtering Method Based on Symmetrical Second Order Systems
Symmetry 2019, 11(6), 813; https://doi.org/10.3390/sym11060813 - 20 Jun 2019
Abstract
This study presents a filtering and sampling structure based on symmetrical second order systems working on half-period. It is shown that undamped second order oscillating systems working on half-period could provide: (i) a large attenuation coefficient for an alternating signal (due to the [...] Read more.
This study presents a filtering and sampling structure based on symmetrical second order systems working on half-period. It is shown that undamped second order oscillating systems working on half-period could provide: (i) a large attenuation coefficient for an alternating signal (due to the filtering second order system), and (ii) a robust sampling procedure (the slope of the generated output being zero at the sampling time moment). Unlike previous studies on the same topics, these results are achieved without the use of an additional integrator. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
Symmetric Networks with Geometric Constraints as Models of Visual Illusions
Symmetry 2019, 11(6), 799; https://doi.org/10.3390/sym11060799 - 16 Jun 2019
Cited by 2
Abstract
Multistable illusions occur when the visual system interprets the same image in two different ways. We model illusions using dynamic systems based on Wilson networks, which detect combinations of levels of attributes of the image. In most examples presented here, the network has [...] Read more.
Multistable illusions occur when the visual system interprets the same image in two different ways. We model illusions using dynamic systems based on Wilson networks, which detect combinations of levels of attributes of the image. In most examples presented here, the network has symmetry, which is vital to the analysis of the dynamics. We assume that the visual system has previously learned that certain combinations are geometrically consistent or inconsistent, and model this knowledge by adding suitable excitatory and inhibitory connections between attribute levels. We first discuss 4-node networks for the Necker cube and the rabbit/duck illusion. The main results analyze a more elaborate model for the Necker cube, a 16-node Wilson network whose nodes represent alternative orientations of specific segments of the image. Symmetric Hopf bifurcation is used to show that a small list of natural local geometric consistency conditions leads to alternation between two global percepts: cubes in two different orientations. The model also predicts brief transitional states in which the percept involves impossible rectangles analogous to the Penrose triangle. A tristable illusion generalizing the Necker cube is modelled in a similar manner. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
Multi-Criteria Decision Making Approach for Hybrid Operation of Wind Farms
Symmetry 2019, 11(5), 675; https://doi.org/10.3390/sym11050675 - 16 May 2019
Cited by 8
Abstract
Hybrid operation of wind farms has been in the limelight in recent years wherein the stochastic nature of wind causes market operators to choose an optimal strategy to maximize profit. The current work deals with a multi-criteria decision making approach to choose the [...] Read more.
Hybrid operation of wind farms has been in the limelight in recent years wherein the stochastic nature of wind causes market operators to choose an optimal strategy to maximize profit. The current work deals with a multi-criteria decision making approach to choose the best possible alternatives for a hybrid wind farm operation. A set of three, non-beneficial criteria, namely wind wakes, wind curtailment, and forced outages, were chosen to evaluate the best alternative. Three methods, (i) Simple Additive Weighting (SAW), (ii) the Technique for Order or Preference by Similarity to Ideal Solution (TOPSIS) and (iii) Complex Proportional Assessment (COPRAS), were applied to identify the best alternative, and the results revealed that for all three methods, borrowing deficit power from a neighboring wind farm is the best alternative. Comparative analyses in terms of the data requirement, the effect of dynamic decision matrices, and rank reversal in wind farm application have also been pioneered. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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Open AccessArticle
Dynamic Modeling and Experiment Research on Twin Ball Screw Feed System Considering the Joint Stiffness
Symmetry 2018, 10(12), 686; https://doi.org/10.3390/sym10120686 - 01 Dec 2018
Cited by 6
Abstract
It is of great significance to study the dynamic characteristics of twin ball screw (TBS) feed system to improve the precision of gantry-type dual-driven computer numerical control (CNC) machine tools. In this paper, an equivalent dynamic model of the TBS feed system is [...] Read more.
It is of great significance to study the dynamic characteristics of twin ball screw (TBS) feed system to improve the precision of gantry-type dual-driven computer numerical control (CNC) machine tools. In this paper, an equivalent dynamic model of the TBS feed system is established utilizing lumped mass method considering the stiffness of joints. Equivalent axial stiffness of screw-nut joints and bearing joints are both calculated by Hertz contact theory. Furthermore, a friction model is proposed because the friction force of the screw nut affects the stiffness of the joints. Then, the friction parameters are obtained by using the nonlinear system identification method. Meanwhile, a finite element model (FEM) is developed to assess the dynamic characteristics of TBS feed system under the stiffness of joints. Finally, validation experiments are conducted, and the results show that the positions of the nut and the velocities of worktable greatly affect the dynamic characteristics of the TBS feed system. Compared with the theoretical calculation, FEM and experiments indicate that the dynamic modeling proposed in this article can reach a higher accuracy. Full article
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
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