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

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 24842

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Prof. Dr. Quanxin Zhu

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Guest Editor

School of Mathematical Sciences and Statistics, Hunan Normal University, Changsha 410081, China
Interests: stochastic control; stochastic systems; stochastic stability; stochastic delayed systems; markovian jump systems; stochastic complex networks
Special Issues, Collections and Topics in MDPI journals

Dr. Fanchao Kong

E-Mail Website
Guest Editor

School of Mathematics and Statistics, Anhui Normal University, Wuhu 241000, China
Interests: the stability of discontinuous systems and applications to the neural networks
Special Issues, Collections and Topics in MDPI journals

Dr. Zuowei Cai

E-Mail Website
Guest Editor

School of Information Science and Engineering, Hunan Women’s University, Changsha 410002, China
Interests: differential inclusions; neural networks; mathematical biology and stability analysis of dynamic systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry intends to publish a Special Issue on the heories and applications related to symmetry/asymmetry. The aim of this Special Issue is to highlight papers that show dynamics, control, optimization, and applications of nonlinear systems. This has recently become an increasingly popular subject, with an impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers that share this objective are especially welcome.

Submit your paper and select the Journal “Symmetry” and the Special Issue “Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications ” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Prof. Dr. Quanxin Zhu
Dr. Fanchao Kong
Dr. Zuowei Cai
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

Stability analysis of discrete and continuous dynamical systems
Nonlinear dynamics in biological complex systems
Stability and stabilization of stochastic systems
Mathematical models in statistics and probability
Synchronization of oscillators and chaotic systems
Optimization methods of complex systems
Reliability modeling and system optimization
Computation and control over networked systems

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Published Papers (14 papers)

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Editorial

Jump to: Research, Review

3 pages, 180 KiB

Open AccessEditorial

Special Issue “Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications”

by Quanxin Zhu, Fanchao Kong and Zuowei Cai

Symmetry 2023, 15(1), 26; https://doi.org/10.3390/sym15010026 - 22 Dec 2022

Cited by 8 | Viewed by 1023

Abstract

Nonlinear systems described by differential equations are of great theoretical significance and do frequently arise in practice [...] Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

Research

Jump to: Editorial, Review

10 pages, 268 KiB

Open AccessArticle

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

by Saleem Khan, Shariefuddin Pirzada and Yilun Shang

Symmetry 2022, 14(9), 1937; https://doi.org/10.3390/sym14091937 - 17 Sep 2022

Cited by 10 | Viewed by 1712

Abstract

The reciprocal distance Laplacian matrix of a connected graph G is defined as

R D^{L} (G) = R T (G) - R D (G)

, where

R T (G)

is the diagonal matrix of [...] Read more.

The reciprocal distance Laplacian matrix of a connected graph G is defined as

R D^{L} (G) = R T (G) - R D (G)

, where

R T (G)

is the diagonal matrix of reciprocal distance degrees and

R D (G)

is the Harary matrix. Clearly,

R D^{L} (G)

is a real symmetric matrix, and we denote its eigenvalues as

λ_{1} (R D^{L} (G)) \geq λ_{2} (R D^{L} (G)) \geq \dots \geq λ_{n} (R D^{L} (G))

. The largest eigenvalue

λ_{1} (R D^{L} (G))

R D^{L} (G)

, denoted by

λ (G)

, is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree

R T_{m a x}

, minimum reciprocal distance degree

R T_{m i n}

, and Harary index

H (G)

of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius

λ (G)

in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread

R D L S (G) = λ_{1} (R D^{L} (G)) - λ_{n - 1} (R D^{L} (G))

in terms of various graph parameters. We determine the extremal graphs in many cases. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

16 pages, 874 KiB

Open AccessArticle

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

by Wafa Shammakh, A. George Maria Selvam, Vignesh Dhakshinamoorthy and Jehad Alzabut

Symmetry 2022, 14(9), 1877; https://doi.org/10.3390/sym14091877 - 8 Sep 2022

Cited by 5 | Viewed by 1361

Abstract

The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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8 pages, 268 KiB

Open AccessArticle

On New Filters in Ordered Semigroups

by Madeleine Al-Tahan, Bijan Davvaz, Ahsan Mahboob, Sarka Hoskova-Mayerova and Alena Vagaská

Symmetry 2022, 14(8), 1564; https://doi.org/10.3390/sym14081564 - 29 Jul 2022

Cited by 4 | Viewed by 1593

Abstract

Ordered semigroups are understood through their subsets. The aim of this article is to study ordered semigroups through some new substructures. In this regard, quasi-filters and

(m, n)

-quasi-filters of ordered semigroups are introduced as new types of filters. Some [...] Read more.

Ordered semigroups are understood through their subsets. The aim of this article is to study ordered semigroups through some new substructures. In this regard, quasi-filters and

(m, n)

-quasi-filters of ordered semigroups are introduced as new types of filters. Some properties of the new concepts are investigated, different examples are constructed, and the relations between quasi-filters and quasi-ideals as well as between

(m, n)

-quasi-filters and

(m, n)

-quasi-ideals are discussed. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

22 pages, 6017 KiB

Open AccessArticle

Hopf Bifurcation and Control of a Fractional-Order Delay Stage Structure Prey-Predator Model with Two Fear Effects and Prey Refuge

by Yongzhong Lan, Jianping Shi and Hui Fang

Symmetry 2022, 14(7), 1408; https://doi.org/10.3390/sym14071408 - 9 Jul 2022

Cited by 8 | Viewed by 1769

Abstract

A generalized delay stage structure prey-predator model with fear effect and prey refuge is considered in this paper via introducing fractional-order and fear effect induced by immature predators. Hopf bifurcation and control of this system are investigated though regarding the delay as the parameter. Firstly, by using the method of linearization and Laplace transform, the roots of the characteristic equation of the linearized system of the original system are discussed, and the sufficient conditions for the system exhibits an unstable state of symmetrical periodic oscillation (Hopf bifurcation) are explored. Secondly, a linear delay feedback controller is added to the system to increase the stability domain successfully. Thirdly, numerical simulations are performed to validate the theoretical analysis, and the various impacts on the dynamical behavior of the system occurring by fear effects, prey refuge, and each fractional-order are illustrated, respectively. Furthermore, the influence of feedback gain on the bifurcation critical point is analyzed. Finally, an analysis based on the results and in-depth research about this system under the biological background is stated in the conclusion. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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13 pages, 305 KiB

Open AccessArticle

Fixed-Time Synchronization Analysis of Genetic Regulatory Network Model with Time-Delay

by Yajun Zhou and You Gao

Symmetry 2022, 14(5), 951; https://doi.org/10.3390/sym14050951 - 7 May 2022

Cited by 1 | Viewed by 1395

Abstract

The synchronous genetic regulatory networks model includes the drive system and response system, and the drive-response system is symmetric. From a biological point of view, this model illustrates the dynamic behaviors in gene-to-protein processes, in terms of transcription and translation. This paper introduces a model of genetic regulatory networks with time delay. The fixed-time synchronization control problem of the proposed model is studied based on fixed-time stability theory and the Lyapunov method. Concretely, the authors first propose a way to switch from the given model to matrix form. Then, two types of novel controllers are designed and the corresponding sufficient conditions are investigated respectively to ensure the fixed-time synchronization goal. Moreover, the settling times of fixed-time synchronization are estimated for the addressed discontinuous controllers, which are not dependent on the initial or delayed states of the model. Finally, numerical simulations are presented and compared to illustrate the benefits of the theoretical results. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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15 pages, 2912 KiB

Open AccessArticle

Vibration Force Suppression of Magnetically Suspended Flywheel Based on Compound Repetitive Control

by Yuan Zeng, Kun Liu, Jingbo Wei and Zhizhou Zhang

Symmetry 2022, 14(5), 949; https://doi.org/10.3390/sym14050949 - 6 May 2022

Cited by 1 | Viewed by 1472

Abstract

To realize the hyperstatic performance index of a magnetically suspended flywheel and simultaneously suppress the vibration force caused by mass imbalance and sensor runout, a compound control method based on a repetitive controller and displacement force compensation of the synchronous force is proposed. First, the mechanism of different interference vibration forces is analyzed by establishing a model of the magnetically suspended flywheel. The analysis shows that the x–y direction is symmetric, and the flywheel structure has symmetry. Second, considering the symmetry of the x- and y-directions, the x-direction is taken as an example for analysis, the parameter design and stability analysis are carried out, and the range of parameters of the compound repetitive control method is obtained. Finally, a flywheel with different speeds is simulated. It was found that the vibration force of each frequency can be suppressed by the compound control method, and the inhibition rate of the vibration force can reach as much as 95%. The results show that the unbalanced vibration and vibration force caused by the sensor runout can be effectively suppressed by using the compound repetitive control method. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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11 pages, 311 KiB

Open AccessArticle

Stochastic Finite-Time Stability for Stochastic Nonlinear Systems with Stochastic Impulses

by Wei Hu

Symmetry 2022, 14(4), 817; https://doi.org/10.3390/sym14040817 - 14 Apr 2022

Cited by 6 | Viewed by 1993

Abstract

In this paper, some novel stochastic finite-time stability criteria for stochastic nonlinear systems with stochastic impulse effects are established. The results in this paper blackgeneralized the related results in from two aspects: 1. the model in is the deterministic systems, which means that the noise effect that can be described as a symmetric Markov process Brownian motion is considered in our models; 2. the stochastic finite-time stability criterion is established in this paper, not the asymptotic stability and the input-to-state stability that are studied in the form literature. Finally, an example is given to show the significance blackand usefulness of our results. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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19 pages, 4032 KiB

Open AccessArticle

Stability Switching Curves and Hopf Bifurcation of a Fractional Predator–Prey System with Two Nonidentical Delays

by Shuangfei Li, Yingxian Zhu, Yunxian Dai and Yiping Lin

Symmetry 2022, 14(4), 643; https://doi.org/10.3390/sym14040643 - 22 Mar 2022

Cited by 7 | Viewed by 1814

Abstract

In this paper, we propose and analyze a three-dimensional fractional predator–prey system with two nonidentical delays. By choosing two delays as the bifurcation parameter, we first calculate the stability switching curves in the delay plane. By judging the direction of the characteristic root across the imaginary axis in stability switching curves, we obtain that the stability of the system changes when two delays cross the stability switching curves, and then, the system appears to have bifurcating periodic solutions near the positive equilibrium, which implies that the trajectory of the system is the axial symmetry. Secondly, we obtain the conditions for the existence of Hopf bifurcation. Finally, we give one example to verify the correctness of the theoretical analysis. In particular, the geometric stability switch criteria are applied to the stability analysis of the fractional differential predator–prey system with two delays for the first time. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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21 pages, 357 KiB

Open AccessArticle

Some New Estimates on Coordinates of Left and Right Convex Interval-Valued Functions Based on Pseudo Order Relation

by Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Kamsing Nonlaopon and Yasser S. Hamed

Symmetry 2022, 14(3), 473; https://doi.org/10.3390/sym14030473 - 25 Feb 2022

Cited by 8 | Viewed by 1434

Abstract

The relevance of convex and non-convex functions in optimization research is well known. Due to the behavior of its definition, the idea of convexity also plays a major role in the subject of inequalities. The main concern of this paper is to establish new integral inequalities for newly defined left and right convex interval-valued function on coordinates through pseudo order relation and double integral. Some of the Hermite–Hadamard type inequalities for the product of two left and right convex interval-valued functions on coordinates are also obtained. Moreover, Hermite–Hadamard–Fejér type inequalities are also derived for left and right convex interval-valued functions on coordinates. Some useful examples are also presented to prove the validity of this study. The proved results of this paper are generalizations of many known results, which are proved by Dragomir, Latif et al. and Zhao, and can be vied as applications of this study. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

17 pages, 663 KiB

Open AccessArticle

On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

by Michal Staš

Symmetry 2021, 13(12), 2441; https://doi.org/10.3390/sym13122441 - 17 Dec 2021

Cited by 10 | Viewed by 2683

Abstract

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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9 pages, 278 KiB

Open AccessArticle

Convergence on Population Dynamics and High-Dimensional Haddock Conjecture

by Wenke Wang, Le Li, Xuejun Yi and Chuangxia Huang

Symmetry 2021, 13(12), 2252; https://doi.org/10.3390/sym13122252 - 26 Nov 2021

Cited by 2 | Viewed by 1379

Abstract

One fundamental step towards grasping the global dynamic structure of a population system involves characterizing the convergence behavior (specifically, how to characterize the convergence behavior). This paper focuses on the neutral functional differential equations arising from population dynamics. With the help of monotonicity [...] Read more.

ω

-limited set and special point. Meanwhile, we prove that every bounded solution converges to a constant vector, as t tends to positive infinity. Our results correlate with the findings from earlier publications, and our proof yields an improved Haddock conjecture. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

18 pages, 479 KiB

Open AccessArticle

Periodicity on Neutral-Type Inertial Neural Networks Incorporating Multiple Delays

by Jian Zhang, Ancheng Chang and Gang Yang

Symmetry 2021, 13(11), 2231; https://doi.org/10.3390/sym13112231 - 22 Nov 2021

Cited by 4 | Viewed by 1797

Abstract

The classical Hopefield neural networks have obvious symmetry, thus the study related to its dynamic behaviors has been widely concerned. This research article is involved with the neutral-type inertial neural networks incorporating multiple delays. By making an appropriate Lyapunov functional, one novel sufficient stability criterion for the existence and global exponential stability of T-periodic solutions on the proposed system is obtained. In addition, an instructive numerical example is arranged to support the present approach. The obtained results broaden the application range of neutral-types inertial neural networks. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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Review

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18 pages, 314 KiB

Open AccessReview

Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges

by Víctor Ayala, Adriano Da Silva and José Ayala

Symmetry 2022, 14(4), 661; https://doi.org/10.3390/sym14040661 - 24 Mar 2022

Cited by 1 | Viewed by 1507

Abstract

The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group G, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on G: linear and invariant vector fields. A linear vector field on G is determined by its flow, a 1-parameter group of

A u t (G)

, the Lie group of G-automorphisms. An invariant vector field is just an element of the Lie algebra g of G. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and G. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper. Full article

(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)

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