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Axioms
Special Issues
Complex Networks and Dynamical Systems
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Complex Networks and Dynamical Systems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 2234

Special Issue Editors

Dr. Shang Gao

E-Mail Website
Guest Editor
College of Science, Northeast Forestry University, Harbin, China
Interests: complex systems; control theory; stochastic differential equations; stability; synchronization
Dr. Chunmei Zhang

E-Mail Website
Guest Editor
Department of Mathematics, Southwest Jiaotong University, Chengdu, China
Interests: networked control system; stochastic differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ying Guo

E-Mail Website
Guest Editor
Faculty of Science, Qingdao University of Technology, Qingdao, China
Interests: network control systems; the dynamic property of stochastic differential equations

Special Issue Information

Dear Colleagues:

With the development of science and technology, complex networks can be seen everywhere in the real world. Examples include the Internet, social networks, neural networks, and many others. In recent years, complex networks have attracted the attention of scholars in many fields including mathematics, biology, engineering and sociology. How to model actual complex networks and study their dynamic behavior is a very interesting and important topic in the field of complex network research. In mathematics, complex networks are usually described by differential or difference equations. In addition, practical applications depend on its dynamic properties such as stability, synchronization, periodicity, etc.

In this Special Issue of Axioms, we aim to explore the latest research and developments in dynamic properties of complex network control systems described by differential or difference equations. We will feature high-quality papers that cover a range of topics, including stability, synchronization and consistency of complex networks.

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

  1. Dynamic behaviors of complex networks in the real world and their applications;
  2. Dynamic properties of different complex dynamical networks.

I look forward to receiving your contributions.

Dr. Shang Gao
Dr. Chunmei Zhang
Prof. Dr. Ying Guo
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. Axioms 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

  • complex networks
  • stability
  • white noise
  • telegraph noise
  • stochastic systems
  • control theory
  • synchronization
  • periodicity

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (4 papers)

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Research

26 pages, 789 KiB  
Article
Stability and Hopf Bifurcation of Fractional-Order Quaternary Numerical Three-Neuron Neural Networks with Different Types of Delays
by Qiankun Wang, Tianzeng Li, Yu Wang and Xiaowen Tan
Axioms 2025, 14(5), 366; https://doi.org/10.3390/axioms14050366 - 13 May 2025
Viewed by 127
Abstract
In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence [...] Read more.
In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence and uniqueness of the solution for the considered fractional-order delayed quaternion neural networks are proven by using the compression mapping theorem. It is demonstrated that the solutions of the involved fractional delayed quaternion neural networks are bounded by constructing appropriate functions. Some sufficient conditions for the stability and Hopf bifurcation of the considered fractional-order delayed quaternion neural networks are established by utilizing the stability theory of fractional differential equations and basic bifurcation knowledge. To validate the rationality of the theoretical results, corresponding simulation results and bifurcation diagrams are provided. The relationship between the order of appearance of bifurcation phenomena and the order is also studied, revealing that bifurcation phenomena occur later as the order increases. The theoretical results established in this paper are of significant guidance for the design and improvement of neural networks. Full article
(This article belongs to the Special Issue Complex Networks and Dynamical Systems)
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24 pages, 2118 KiB  
Article
New μ-Synchronization Criteria for Nonlinear Drive–Response Complex Networks with Uncertain Inner Couplings and Variable Delays of Unknown Bounds
by Anran Zhou, Chongming Yang, Chengbo Yi and Hongguang Fan
Axioms 2025, 14(3), 161; https://doi.org/10.3390/axioms14030161 - 23 Feb 2025
Viewed by 286
Abstract
Since the research of μ-synchronization helps to explore how complex networks (CNs) work together to produce complex behaviors, the μ-synchronization task for uncertain time-delayed CNs is studied in our work. Especially, bounded external perturbations and variable delays of unknown bounds containing [...] Read more.
Since the research of μ-synchronization helps to explore how complex networks (CNs) work together to produce complex behaviors, the μ-synchronization task for uncertain time-delayed CNs is studied in our work. Especially, bounded external perturbations and variable delays of unknown bounds containing coupling delays, internal delays, and pulse delays are all taken into consideration, making the model more general. Through the μ-stable theory together with the hybrid impulsive control technique, the problems caused by uncertain inner couplings, time-varying delays, and perturbations can be solved, and novel synchronization criteria are gained for the μ-synchronization of the considered CNs. Different from traditional models, it is not necessary for the coupling matrices to meet the zero-row-sum condition, and the control protocol relaxes the constraint of time delays on impulse intervals. Moreover, numerical experiments and image encryption algorithms are carried out to verify our theoretical results’ effectiveness. Full article
(This article belongs to the Special Issue Complex Networks and Dynamical Systems)
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13 pages, 517 KiB  
Article
First and Second Integrals of Hopf–Langford-Type Systems
by Vassil M. Vassilev and Svetoslav G. Nikolov
Axioms 2025, 14(1), 8; https://doi.org/10.3390/axioms14010008 - 27 Dec 2024
Viewed by 675
Abstract
The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a [...] Read more.
The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable. Full article
(This article belongs to the Special Issue Complex Networks and Dynamical Systems)
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14 pages, 375 KiB  
Article
The Stability of a Predator–Prey Model with Cross-Dispersal in a Multi-Patch Environment
by Keyao Xu, Keyu Peng and Shang Gao
Axioms 2024, 13(11), 783; https://doi.org/10.3390/axioms13110783 - 13 Nov 2024
Viewed by 675
Abstract
This paper investigates the stability of predator–prey models within multi-patch environments, with a particular focus on the influence of cross-dispersion across patches. We apply Kirchhoff’s matrix tree theorem and Liapunov’s method to derive criteria related to the cross-dispersion topology, thus solving the challenge [...] Read more.
This paper investigates the stability of predator–prey models within multi-patch environments, with a particular focus on the influence of cross-dispersion across patches. We apply Kirchhoff’s matrix tree theorem and Liapunov’s method to derive criteria related to the cross-dispersion topology, thus solving the challenge of determining global asymptotic stability conditions. The method incorporates realistic ecological interactions and spatial heterogeneity, offering a framework for stability analysis. Our findings demonstrate that an appropriate level of cross-dispersion can effectively mitigate oscillations and foster convergence toward equilibrium. Two numerical examples validate these theoretical results and demonstrate the feasibility and effectiveness of the model across multiple patches. Full article
(This article belongs to the Special Issue Complex Networks and Dynamical Systems)
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