Advances in Fractional-Order Neural Networks, Volume II

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

Deadline for manuscript submissions: closed (20 April 2023) | Viewed by 15045

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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
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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

Special Issue Information

Dear Colleagues,

Fractional-order neural network models have become an active research subject and have attracted increasing 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. On the other side, challenges and knowledge from research in science and engineering motivate new advancements in the area of fractional-order neural networks.

Following the successful production of Volume I of this Special Issue, we are pleased to 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
Dr. Gani Stamov
Prof. Dr. Xiaodi Li
Guest Editors

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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.

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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
  • modeling qualitative theory (stability, periodicity, almost periodicity, oscillation theory)
  • control stabilization applications to real-world phenomena

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Published Papers (13 papers)

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Editorial

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3 pages, 200 KiB  
Editorial
Advances in Fractional-Order Neural Networks, Volume II
by Ivanka Stamova, Gani Stamov and Xiaodi Li
Fractal Fract. 2023, 7(12), 845; https://doi.org/10.3390/fractalfract7120845 - 29 Nov 2023
Viewed by 909
Abstract
Fractional-order neural network models have become an active research subject and have attracted increasing attention in many fields [...] Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)

Research

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20 pages, 543 KiB  
Article
Finite-Time Synchronization for Stochastic Fractional-Order Memristive BAM Neural Networks with Multiple Delays
by Lili Chen, Minghao Gong, Yanfeng Zhao and Xin Liu
Fractal Fract. 2023, 7(9), 678; https://doi.org/10.3390/fractalfract7090678 - 10 Sep 2023
Cited by 1 | Viewed by 737
Abstract
This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison [...] Read more.
This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison principle. This criterion provides a new approach to analyze the finite-time synchronization problem of neural networks with stochasticity. Finally, numerical simulations are provided to demonstrate the effectiveness and superiority of the obtained results. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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16 pages, 6869 KiB  
Article
Dynamical Analysis of the Incommensurate Fractional-Order Hopfield Neural Network System and Its Digital Circuit Realization
by Miao Wang, Yuru Wang and Ran Chu
Fractal Fract. 2023, 7(6), 474; https://doi.org/10.3390/fractalfract7060474 - 15 Jun 2023
Cited by 2 | Viewed by 916
Abstract
Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of [...] Read more.
Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q0, q1, and q2. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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23 pages, 868 KiB  
Article
Fractional-Order Impulsive Delayed Reaction-Diffusion Gene Regulatory Networks: Almost Periodic Solutions
by Trayan Stamov, Gani Stamov and Ivanka Stamova
Fractal Fract. 2023, 7(5), 384; https://doi.org/10.3390/fractalfract7050384 - 03 May 2023
Cited by 3 | Viewed by 1074
Abstract
The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time are considered. New almost periodic and perfect Mittag–Leffler stability criteria are [...] Read more.
The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time are considered. New almost periodic and perfect Mittag–Leffler stability criteria are proposed. Lyapunov’s like impulsive functions, the properties of the fractional derivatives and comparison principle are the main tools in the investigation. Illustrative examples are also presented to demonstrate the proposed criteria. Our results contribute to the development of qualitative the theory of fractional-order gene regulatory networks. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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14 pages, 6002 KiB  
Article
A Model-Free Finite-Time Control Technique for Synchronization of Variable-Order Fractional Hopfield-like Neural Network
by Fawaz W. Alsaade, Mohammed S. Al-zahrani, Qijia Yao and Hadi Jahanshahi
Fractal Fract. 2023, 7(5), 349; https://doi.org/10.3390/fractalfract7050349 - 25 Apr 2023
Cited by 2 | Viewed by 1040
Abstract
Although the literature presents promising techniques for the control of integer-order systems, control and synchronizing fractional systems still need further improvement to ensure their robustness and convergence time. This study aims to address this issue by proposing a model-free and finite-time super-twisting control [...] Read more.
Although the literature presents promising techniques for the control of integer-order systems, control and synchronizing fractional systems still need further improvement to ensure their robustness and convergence time. This study aims to address this issue by proposing a model-free and finite-time super-twisting control technique for a variable-order fractional Hopfield-like neural network. The proposed controller is enhanced with an intelligent observer to account for disturbances and uncertainties in the chaotic model of the Hopfield-like neural network. The controller is able to regulate the system even when its complex variable-order fractional dynamic is completely unknown. Moreover, the proposed technique guarantees finite-time convergence of the closed-loop system. First, the dynamics of the variable-order fractional Hopfield-like neural network are examined. Then, the control design is described and its finite-time stability is proven. The controller is then applied to the variable-order fractional system and tested under two different scenarios to evaluate its performance. The results of the simulations demonstrate the excellent performance of the proposed method in both scenarios. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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27 pages, 618 KiB  
Article
Enhancing the Mathematical Theory of Nabla Tempered Fractional Calculus: Several Useful Equations
by Yiheng Wei, Linlin Zhao, Xuan Zhao and Jinde Cao
Fractal Fract. 2023, 7(4), 330; https://doi.org/10.3390/fractalfract7040330 - 14 Apr 2023
Cited by 2 | Viewed by 1041
Abstract
Although many applications of fractional calculus have been reported in literature, modeling the physical world using this technique is still a challenge. One of the main difficulties in solving this problem is that the long memory property is necessary, whereas the infinite memory [...] Read more.
Although many applications of fractional calculus have been reported in literature, modeling the physical world using this technique is still a challenge. One of the main difficulties in solving this problem is that the long memory property is necessary, whereas the infinite memory is undesirable. To address this challenge, a new type of nabla fractional calculus with a weight function is formulated, which combines the benefits of nabla fractional calculus and its tempered counterpart, making it highly valuable for modeling practical systems. However, many properties of this calculus are still unclear and need to be discovered. Therefore, this paper gives particular emphasis to the topic, developing some remarkable properties, i.e., the equivalence relation, the nabla Taylor formula, and the nabla Laplace transform of such nabla tempered fractional calculus. All the developed properties greatly enrich the mathematical theory of nabla tempered fractional calculus and provide high value and potential for further applications. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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18 pages, 1117 KiB  
Article
Dynamic Event-Triggered Consensus for Fractional-Order Multi-Agent Systems without Intergroup Balance Condition
by Bingrui Xu and Bing Li
Fractal Fract. 2023, 7(3), 268; https://doi.org/10.3390/fractalfract7030268 - 17 Mar 2023
Cited by 1 | Viewed by 934
Abstract
This paper deals with the problem of group consensus for a fractional-order multi-agent system (FOMAS) without considering the intergroup balance condition. By adopting a dynamic event-triggered mechanism, the updating frequency of control input is significantly reduced while the consensus performance is maintained. By [...] Read more.
This paper deals with the problem of group consensus for a fractional-order multi-agent system (FOMAS) without considering the intergroup balance condition. By adopting a dynamic event-triggered mechanism, the updating frequency of control input is significantly reduced while the consensus performance is maintained. By utilizing the Lyapunov direct method and the properties of a fractional-order derivative, several novel criteria are presented for analyzing the Mittag–Leffler stability of error systems and excluding the Zeno behavior in the triggering mechanism. An example and its simulations are demonstrated to prove the validity of the theoretical results. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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13 pages, 2182 KiB  
Article
Non-Periodicity of Complex Caputo Like Fractional Differences
by Michal Fečkan and Marius-F. Danca
Fractal Fract. 2023, 7(1), 68; https://doi.org/10.3390/fractalfract7010068 - 07 Jan 2023
Cited by 2 | Viewed by 951
Abstract
Aspects related to non-periodicity of a class of complex maps defined in the sense of Caputo like fractional differences and to the asymptotical stability of fixed points are considered. As example the Mandelbrot map of fractional order is considered. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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14 pages, 1114 KiB  
Article
New Results on Robust Synchronization for Memristive Neural Networks with Fractional Derivatives via Linear Matrix Inequality
by Chao Song, Jinde Cao and Mahmoud Abdel-Aty
Fractal Fract. 2022, 6(10), 585; https://doi.org/10.3390/fractalfract6100585 - 12 Oct 2022
Cited by 3 | Viewed by 1141 | Correction
Abstract
This article mainly concentrates on the synchronization problem for a more general kind of the master–slave memristor-based neural networks with fractional derivative. By applying a continuous-frequency-distributed equivalent model tool, some new outcomes and sufficient conditions on the robust synchronization of the master–slave neural [...] Read more.
This article mainly concentrates on the synchronization problem for a more general kind of the master–slave memristor-based neural networks with fractional derivative. By applying a continuous-frequency-distributed equivalent model tool, some new outcomes and sufficient conditions on the robust synchronization of the master–slave neural networks with uncertainty are proposed via linear matrix inequality (LMI). Finally, two memristive neural networks model with fractional derivatives are presented to validate the efficiency of the theoretical results. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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13 pages, 741 KiB  
Article
The Effect of Caputo Fractional Variable Difference Operator on a Discrete-Time Hopfield Neural Network with Non-Commensurate Order
by Rabia Chaimaà Karoun, Adel Ouannas, Mohammed Al Horani and Giuseppe Grassi
Fractal Fract. 2022, 6(10), 575; https://doi.org/10.3390/fractalfract6100575 - 09 Oct 2022
Cited by 9 | Viewed by 1259
Abstract
In this work, we recall some definitions on fractional calculus with discrete-time. Then, we introduce a discrete-time Hopfield neural network (D.T.H.N.N) with non-commensurate fractional variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that [...] Read more.
In this work, we recall some definitions on fractional calculus with discrete-time. Then, we introduce a discrete-time Hopfield neural network (D.T.H.N.N) with non-commensurate fractional variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that the proposed discrete time Hopfield neural network with non-commensurate fractional variable order has chaotic behavior. Furthermore, we use the 0-1 test and C0 complexity algorithm to confirm and prove the results obtained about the presence of chaos. Finally, simulations are carried out in Matlab to illustrate the results. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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14 pages, 318 KiB  
Article
Split-Plot Designs with Few Whole Plot Factors Containing Clear Effects
by Yuna Zhao
Fractal Fract. 2022, 6(8), 453; https://doi.org/10.3390/fractalfract6080453 - 20 Aug 2022
Cited by 1 | Viewed by 1212
Abstract
Fractional factorial split-plot designs are widely used when it is impractical to perform fractional factorial experiments in a completely random order. When there are too many subplots per whole plot, or too few whole plots, fractional factorial split-plot designs with replicated settings of [...] Read more.
Fractional factorial split-plot designs are widely used when it is impractical to perform fractional factorial experiments in a completely random order. When there are too many subplots per whole plot, or too few whole plots, fractional factorial split-plot designs with replicated settings of the whole plot factors are preferred. However, such an important study is undeveloped in the literature. This paper considers fractional factorial split-plot designs with replicated settings of the WP factors from the viewpoint of clear effects. We investigate the sufficient and necessary conditions for this class of designs to have clear effects. An algorithm is proposed to generate the desirable designs which have the most clear effects of interest. The fractional factorial split-plot design with replicated settings of the WP factors is analysed and the results are discussed. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
18 pages, 978 KiB  
Article
Asymptotic Synchronization of Fractional-Order Complex Dynamical Networks with Different Structures and Parameter Uncertainties
by Xiliang He, Tianzeng Li and Dehui Liu
Fractal Fract. 2022, 6(8), 441; https://doi.org/10.3390/fractalfract6080441 - 14 Aug 2022
Cited by 1 | Viewed by 1257
Abstract
This paper deals with the asymptotic synchronization of fractional-order complex dynamical networks with different structures and parameter uncertainties (FCDNDP). Firstly, the FCDNDP model is proposed by the Riemann–Liouville (R-L) fractional derivative. According to the property of fractional calculus and the Lyapunov direct method, [...] Read more.
This paper deals with the asymptotic synchronization of fractional-order complex dynamical networks with different structures and parameter uncertainties (FCDNDP). Firstly, the FCDNDP model is proposed by the Riemann–Liouville (R-L) fractional derivative. According to the property of fractional calculus and the Lyapunov direct method, an original controller is proposed to achieve the asymptotic synchronization of FCDNDP. Our controller is more adaptable and effective than those in other literature. Secondly, a sufficient condition is given for the asymptotic synchronization of FCDNDP based on the asymptotic stability theorem and the matrix inequality technique. Finally, the numerical simulations verify the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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Other

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18 pages, 922 KiB  
Brief Report
Synchronization in Finite Time of Fractional-Order Complex-Valued Delayed Gene Regulatory Networks
by Lu Wang, Xujun Yang, Hongjun Liu and Xiaofeng Chen
Fractal Fract. 2023, 7(5), 347; https://doi.org/10.3390/fractalfract7050347 - 23 Apr 2023
Cited by 2 | Viewed by 767
Abstract
The synchronization in finite time of fractional-order complex-valued gene networks with time delays is studied in this paper. Several sufficient conditions of the synchronization in finite time for the relevant network models are explored based on feedback controllers and adaptive controllers. Then, the [...] Read more.
The synchronization in finite time of fractional-order complex-valued gene networks with time delays is studied in this paper. Several sufficient conditions of the synchronization in finite time for the relevant network models are explored based on feedback controllers and adaptive controllers. Then, the setting time of the response is estimated by the theory of fractional calculus. Finally, to validate the theoretical results, a numerical example is presented using the proposed two controllers, showing that the setting time based on the adaptive controller is shorter than the that based on the feedback controller. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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