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Fractal Fract
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Frontiers in Fractional-Order Neural Networks
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Frontiers in Fractional-Order Neural Networks

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (20 November 2021) | Viewed by 27494

Special Issue Editors

Dr. Ivanka Stamova

E-Mail Website
Guest Editor
Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
Interests: nonlinear analysis; mathematical modeling; fractional order systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Xiaodi Li

grade E-Mail Website
Guest Editor
School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China
Interests: impulsive control theory; hybrid systems; time-delay systems; neural networks and applied mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Gani Stamov

E-Mail Website
Guest Editor
Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA
Interests: applied mathematics; dynamical systems; differential equations; qualitative properties (almost periodicity, invariant manifolds, asymptotic properties, stability); impulsive perturbations; delays; fractional differential equations; neural networks; economic models
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional-order neural network models have become an active research subject, attracting great attention in many fields. For instance, fractional-order neural networks are recognized as effective tools for modeling, validation and guaranteed learning of dynamical processes in biology, biochemistry, neurocomputing, engineering, physics, economics, etc. Advances in fractional calculus lead to the development of new fractional-order neural network models. Conversely, challenges and knowledge from the research in science and engineering motivate new advancements in the area of fractional-order neural networks.

We invite investigators to contribute original research articles as well as review articles focused on the latest achievements in modeling, control and applications of fractional-order neural networks.

Dr. Ivanka Stamova
Prof. Dr. Xiaodi Li
Dr. Gani Stamov
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. Fractal and Fractional 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 2700 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

  • Fractional Cellular neural networks
  • Fractional Hopfield neural networks
  • Fractional Bidirectional Associate Memory neural network
  • Fractional neural networks with reaction-diffusion terms
  • Fractional Lotka-Volterra neural networks
  • Fractional Cohen-Grossberg neural networks
  • Fractional gene regulatory neural networks
  • Delayed fractional neural network models
  • Impulsive fractional neural Models
  • Uncertain fractional neural networks
  • Modelling
  • Qualitative theory (stability, periodicity, almost periodicity, oscillation theory)
  • Control
  • Stabilization
  • Applications to real-world phenomena
  • Physical informed neural networks (PINN)

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Related Special Issue

Published Papers (12 papers)

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Research

13 pages, 813 KiB  
Article
Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model
by Oana Brandibur and Eva Kaslik
Fractal Fract. 2022, 6(5), 257; https://doi.org/10.3390/fractalfract6050257 - 7 May 2022
Cited by 6 | Viewed by 1953
Abstract
The aim of this work is to describe the dynamics of a fractional-order coupled FitzHugh–Nagumo neuronal model. The equilibrium states are analyzed in terms of their stability properties, both dependently and independently of the fractional orders of the Caputo derivatives, based on recently [...] Read more.
The aim of this work is to describe the dynamics of a fractional-order coupled FitzHugh–Nagumo neuronal model. The equilibrium states are analyzed in terms of their stability properties, both dependently and independently of the fractional orders of the Caputo derivatives, based on recently established theoretical results. Numerical simulations are shown to clarify and exemplify the theoretical results. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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20 pages, 323 KiB  
Article
Global Exponential Stability of Fractional Order Complex-Valued Neural Networks with Leakage Delay and Mixed Time Varying Delays
by M. Hymavathi, G. Muhiuddin, M. Syed Ali, Jehad F. Al-Amri, Nallappan Gunasekaran and R. Vadivel
Fractal Fract. 2022, 6(3), 140; https://doi.org/10.3390/fractalfract6030140 - 2 Mar 2022
Cited by 11 | Viewed by 2515
Abstract
This paper investigates the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov-functional we established sufficient conditions to ensure global exponential stability of the fractional order complex-valued neural networks. The [...] Read more.
This paper investigates the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov-functional we established sufficient conditions to ensure global exponential stability of the fractional order complex-valued neural networks. The stability conditions are established in terms of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
20 pages, 330 KiB  
Article
Robust Stability of Fractional Order Memristive BAM Neural Networks with Mixed and Additive Time Varying Delays
by Xiuping Han, M. Hymavathi, Sumaya Sanober, Bhawna Dhupia and M. Syed Ali
Fractal Fract. 2022, 6(2), 62; https://doi.org/10.3390/fractalfract6020062 - 25 Jan 2022
Cited by 7 | Viewed by 2147
Abstract
This paper is concerned with the problem of the robust stability of fractional-order memristive bidirectional associative memory (BAM) neural networks. Based on Lyapunov theory, fractional-order differential inequalities and linear matrix inequalities (LMI) are applied to obtain a robust asymptotical stability. Finally, numerical examples [...] Read more.
This paper is concerned with the problem of the robust stability of fractional-order memristive bidirectional associative memory (BAM) neural networks. Based on Lyapunov theory, fractional-order differential inequalities and linear matrix inequalities (LMI) are applied to obtain a robust asymptotical stability. Finally, numerical examples are presented. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
17 pages, 319 KiB  
Article
Synchronization of Fractional Order Uncertain BAM Competitive Neural Networks
by M. Syed Ali, M. Hymavathi, Syeda Asma Kauser, Grienggrai Rajchakit, Porpattama Hammachukiattikul and Nattakan Boonsatit
Fractal Fract. 2022, 6(1), 14; https://doi.org/10.3390/fractalfract6010014 - 29 Dec 2021
Cited by 15 | Viewed by 2155
Abstract
This article examines the drive-response synchronization of a class of fractional order uncertain BAM (Bidirectional Associative Memory) competitive neural networks. By using the differential inclusions theory, and constructing a proper Lyapunov-Krasovskii functional, novel sufficient conditions are obtained to achieve global asymptotic stability of [...] Read more.
This article examines the drive-response synchronization of a class of fractional order uncertain BAM (Bidirectional Associative Memory) competitive neural networks. By using the differential inclusions theory, and constructing a proper Lyapunov-Krasovskii functional, novel sufficient conditions are obtained to achieve global asymptotic stability of fractional order uncertain BAM competitive neural networks. This novel approach is based on the linear matrix inequality (LMI) technique and the derived conditions are easy to verify via the LMI toolbox. Moreover, numerical examples are presented to show the feasibility and effectiveness of the theoretical results. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
24 pages, 4562 KiB  
Article
Lower and Upper Bounds of Fractional Metric Dimension of Connected Networks
by Muhammad Javaid, Muhammad Kamran Aslam, Muhammad Imran Asjad, Bander N. Almutairi, Mustafa Inc and Bandar Almohsen
Fractal Fract. 2021, 5(4), 276; https://doi.org/10.3390/fractalfract5040276 - 15 Dec 2021
Viewed by 2069
Abstract
The distance centric parameter in the theory of networks called by metric dimension plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, [...] Read more.
The distance centric parameter in the theory of networks called by metric dimension plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, pattern recognition, integer programming, optimal transportation models and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, lesser number of the utilized nodes, and to characterize the chemical compounds, having unique presentations in molecular networks. After the arrival of its weighted version, known as fractional metric dimension, the rectification of distance-related problems in the aforementioned fields has revived to a great extent. In this article, we compute fractional as well as local fractional metric dimensions of web-related networks called by subdivided QCL, 2-faced web, 3-faced web, and antiprism web networks. Moreover, we analyse their final results using 2D and 3D plots. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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15 pages, 619 KiB  
Article
Quasi-Projective Synchronization of Distributed-Order Recurrent Neural Networks
by Xiao Liu, Kelin Li, Qiankun Song and Xujun Yang
Fractal Fract. 2021, 5(4), 260; https://doi.org/10.3390/fractalfract5040260 - 6 Dec 2021
Cited by 5 | Viewed by 2370
Abstract
In this paper, the quasi-projective synchronization of distributed-order recurrent neural networks is investigated. Firstly, based on the definition of the distributed-order derivative and metric space theory, two distributed-order differential inequalities are obtained. Then, by employing the Lyapunov method, Laplace transform, Laplace final value [...] Read more.
In this paper, the quasi-projective synchronization of distributed-order recurrent neural networks is investigated. Firstly, based on the definition of the distributed-order derivative and metric space theory, two distributed-order differential inequalities are obtained. Then, by employing the Lyapunov method, Laplace transform, Laplace final value theorem, and some inequality techniques, the quasi-projective synchronization sufficient conditions for distributed-order recurrent neural networks are established in cases of feedback control and hybrid control schemes, respectively. Finally, two numerical examples are given to verify the effectiveness of the theoretical results. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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23 pages, 404 KiB  
Article
Integral Representation of the Solutions for Neutral Linear Fractional System with Distributed Delays
by Hristo Kiskinov, Ekaterina Madamlieva, Magdalena Veselinova and Andrey Zahariev
Fractal Fract. 2021, 5(4), 222; https://doi.org/10.3390/fractalfract5040222 - 15 Nov 2021
Cited by 8 | Viewed by 1526
Abstract
In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the [...] Read more.
In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the case of initial functions with first kind discontinuities. This result allows to prove that the corresponding homogeneous system possesses a fundamental matrix C(t,s) continuous in t,t∈[a,∞),a∈R. As an application, integral representations of the solutions of the Cauchy problem for the considered inhomogeneous systems are obtained. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
19 pages, 1829 KiB  
Article
Demand Response Optimal Dispatch and Control of TCL and PEV Agents with Renewable Energies
by Jianqiang Hu and Jinde Cao
Fractal Fract. 2021, 5(4), 140; https://doi.org/10.3390/fractalfract5040140 - 27 Sep 2021
Cited by 4 | Viewed by 1943
Abstract
Demand response (DR) flexible loads can provide fast regulation and ancillary services as reserve capacity in power systems. This paper proposes a demand response optimization dispatch control strategy for flexible thermostatically controlled loads (TCLs) and plug-in electric vehicles (PEVs) with stochastic renewable power [...] Read more.
Demand response (DR) flexible loads can provide fast regulation and ancillary services as reserve capacity in power systems. This paper proposes a demand response optimization dispatch control strategy for flexible thermostatically controlled loads (TCLs) and plug-in electric vehicles (PEVs) with stochastic renewable power injection. Firstly, a chance constraint look-ahead programming model is proposed to maximize the social welfare of both units and load agents, through which the optimal power scheduling for TCL/PEV agents can be obtained. Secondly, two demand response control algorithms for TCLs and PEVs are proposed, respectively, based on the aggregate control models of the load agents. The TCLs are controlled by its temperature setpoints and PEVs are controlled by its charging power such that the DR control objective can be fulfilled. It has been shown that the proposed dispatch and control strategy can coordinate the flexible load agents and the renewable power injection. Finally, the simulation results on a modified IEEE 39 bus system demonstrate the effectiveness of the proposed demand response strategy. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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15 pages, 331 KiB  
Article
Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint
by Fangyuan Wang, Xiaodi Li and Zhaojie Zhou
Fractal Fract. 2021, 5(3), 102; https://doi.org/10.3390/fractalfract5030102 - 24 Aug 2021
Cited by 2 | Viewed by 1736
Abstract
In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and [...] Read more.
In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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18 pages, 329 KiB  
Article
New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces
by Daliang Zhao
Fractal Fract. 2021, 5(3), 89; https://doi.org/10.3390/fractalfract5030089 - 4 Aug 2021
Cited by 4 | Viewed by 1698
Abstract
The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered [...] Read more.
The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered dynamic systems. To conquer the difficulties arising from time delay, we also introduce a suitable delay item in a special complete space. In this work, a nonlinear item is not assumed to have Lipschitz continuity or other growth hypotheses compared with most existing articles. Our main tools are resolvent operator theory and fixed point theory. At last, an example is presented to explain our abstract conclusions. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
17 pages, 354 KiB  
Article
Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis
by Ivanka Stamova, Sotir Sotirov, Evdokia Sotirova and Gani Stamov
Fractal Fract. 2021, 5(3), 78; https://doi.org/10.3390/fractalfract5030078 - 27 Jul 2021
Cited by 11 | Viewed by 2240
Abstract
In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at fixed moments are considered. Criteria for the [...] Read more.
In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at fixed moments are considered. Criteria for the existence and uniqueness of almost periodic waves are proposed. Furthermore, the global perfect Mittag–Leffler stability notion for the almost periodic solution is defined and studied. In addition, a robust global perfect Mittag–Leffler stability analysis is proposed. Lyapunov-type functions and fractional inequalities are applied in the proof. Since the type of Cohen–Grossberg neural networks generalizes several basic neural network models, this research contributes to the development of the investigations on numerous fractional neural network models. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
15 pages, 1692 KiB  
Article
Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games
by Zhangcheng Feng, Wenying Xu and Jinde Cao
Fractal Fract. 2021, 5(3), 62; https://doi.org/10.3390/fractalfract5030062 - 28 Jun 2021
Cited by 2 | Viewed by 2363
Abstract
This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence [...] Read more.
This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm. Full article
(This article belongs to the Special Issue Frontiers in Fractional-Order Neural Networks)
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