Advances in Fractional-Order Neural Networks, Volume II
1
Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
2
School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2023, 7(12), 845; https://doi.org/10.3390/fractalfract7120845
Submission received: 21 November 2023
/
Accepted: 27 November 2023
/
Published: 29 November 2023
(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)
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 the modeling, validation and guaranteed learning of dynamical processes in biology, biochemistry, neurocomputing, engineering, physics, economics, etc. [1,2,3,4,5]. Advances in fractional calculus lead to the development of new fractional-order neural network models. From the other side, challenges and knowledge from research in science and engineering motivate new advancements in the area of fractional-order neural networks [6,7,8,9].
After the successful production of Volume I of this Special Issue, we invited investigators to contribute original research articles as well as review articles focused on the latest achievements in the modeling, control and applications of fractional-order neural networks.
Volume II of the Special Issue was successful; twelve research papers were published that addressed advances in fractional-order neural networks.
Four papers addressed interesting topics related to the synchronization of fractional-order neural networks, which is one of the significant research directions in such systems.
For example, Wang et al. studied the synchronization in finite time of fractional-order complex-valued gene networks with time delays. They established several synchronization in finite time criteria based on feedback controllers and adaptive controllers. The setting time of the response is estimated using the theory of fractional calculus.
He, Li and Liu investigated the asymptotic synchronization of fractional-order complex dynamical networks with different structures and parameter uncertainties. The proposed controller is more adaptable and effective and, as such, the derived results extend some of those found in the existing literature. The validity and feasibility of the theoretical outcomes are confirmed via two simulation instances.
Song, Cao and Abdel-Aty considered the synchronization problem for a more general kind of master–slave memristor-based neural networks with fractional derivatives. 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.
Chen et al. studied the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks with discontinuous jumps. They proposed a novel criterion for finite-time synchronization by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison principle. This criterion provides a new approach to analyzing the finite-time synchronization problem of neural networks with stochasticity.
Variable-order fractionals are a relatively recent development in the field of fractional neural networks. This extremely interesting topic has been investigated in two papers.
Karoun et al. introduced a discrete-time Hopfield neural network with non-commensurate fractional variable orders for three neurons. Its chaotic behavior was studied via phase-plot portraits, bifurcation and Lyapunov exponent diagrams.
Alsaade et al. proposed a model-free and finite-time super-twisting control technique for a variable-order fractional Hopfield-like neural network. The controller proposed in their research is able to regulate the system even when its complex variable-order fractional dynamic is completely unknown.
One of the most important concepts in neural network models in a periodic environment is that of periodicity. However, in real-world problems, the exact periodicity of the states is usually too strong and has limited applicability.
Feckan and Danca considered some aspects related to the non-periodicity of a class of complex maps defined in the sense of Caputo-like fractional differences and related to the asymptotical stability of fixed points. The presented results are exemplified in the case of the Mandelbrot set of fractional order.
Stamov et al. studied the almost periodic behavior of factional-order impulsive delayed reaction–diffusion gene regulatory networks with Caputo-type fractional-order derivatives and impulsive disturbances at unfixed instants of time. In addition, using Lyapunov-like impulsive functions, perfect Mittag–Leffler stability criteria were proposed.
Very interesting results related to fractional factorial split-plot designs, with replicated settings of the whole plot factors from the viewpoint of clear effects, are proposed by Zhao.
Xu and Li studied the problem of a group consensus for a fractional-order multi-agent system without considering the intergroup balance condition. By adopting a dynamic event-triggered mechanism, the updating frequency of the control input is significantly reduced while the consensus performance is maintained.
The study by Wei et al. is devoted to one of the main challenges in using a fractional-order neural network modeling approach, namely the fact that a long memory property is necessary, whereas infinite memory is undesirable. Their study puts particular emphasis on the topic, developing some remarkable properties such as the equivalence relation, the nabla Taylor formula and the nabla Laplace transform of such nabla-tempered fractional calculus.
Finally, the paper authored by Wang, Wang and Chu investigated Hopfield-type neural networks with fractional derivatives of incommensurate fractional orders. Numerous fundamental and qualitative properties were studied, including dissipative properties and stabilization.
The editors of this Special Issue “Advances in Fractional-Order Neural Networks, Volume II” would like to express their sincere gratitude to all the authors who contributed their valuable works, and also to the reviewers for their exceptional efforts in reviewing the manuscripts. We believe that the selected papers will enrich the readers’ knowledge and will stimulate the continuing efforts to develop the theory and applications of fractional-order neural networks, which will continue to be one of the dominant themes in mathematics and mathematics applications due to its theoretical and practical significance.
Conflicts of Interest
The authors declare no conflict of interest.
List of Contributions
- Wang, L.; Yang, X.; Liu, H.; Chen, X. Synchronization in Finite Time of Fractional-Order Complex-Valued Delayed Gene Regulatory Networks. Fractal Fract. 2023, 7, 347. https://doi.org/10.3390/fractalfract7050347.
- He, X.; Li, T.; Liu, D. Asymptotic Synchronization of Fractional-Order Complex Dynamical Networks with Different Structures and Parameter Uncertainties. Fractal Fract. 2022, 6, 441. https://doi.org/10.3390/fractalfract6080441.
- Song, C.; Cao, J.; Abdel-Aty, M. New Results on Robust Synchronization for Memristive Neural Networks with Fractional Derivatives via Linear Matrix Inequality. Fractal Fract. 2022, 6, 585. https://doi.org/10.3390/fractalfract6100585.
- Chen, L.; Gong, M.; Zhao, Y.; Liu, X. Finite-Time Synchronization for Stochastic Fractional-Order Memristive BAM Neural Networks with Multiple Delays. Fractal Fract. 2023, 7, 678. https://doi.org/10.3390/fractalfract7090678.
- Karoun, R.C.; Ouannas, A.; Horani, M.A.; Grassi, G. The Effect of Caputo Fractional Variable Difference Operator on a Discrete-Time Hopfield Neural Network with Non-Commensurate Order. Fractal Fract. 2022, 6, 575. https://doi.org/10.3390/fractalfract6100575.
- Alsaade, F.W.; Al-zahrani, M.S.; Yao, Q.; Jahanshahi, H. A Model-Free Finite-Time Control Technique for Synchronization of Variable-Order Fractional Hopfield-like Neural Network. Fractal Fract. 2023, 7, 349. https://doi.org/10.3390/fractalfract7050349.
- Feckan, M.; Danca, M.-F. Non-Periodicity of Complex Caputo Like Fractional Differences. Fractal Fract. 2023, 7, 68. https://doi.org/10.3390/fractalfract701006.
- Stamov, T.; Stamov, G.; Stamova, I. Fractional-Order Impulsive Delayed Reaction-Diffusion Gene Regulatory Networks: Almost Periodic Solutions. Fractal Fract. 2023, 7, 384. https://doi.org/10.3390/fractalfract7050384.
- Zhao, Y. Split-Plot Designs with Few Whole Plot Factors Containing Clear Effects. Fractal Fract. 2022, 6, 453. https://doi.org/10.3390/fractalfract6080453.
- Xu, B.; Li, B. Dynamic Event-Triggered Consensus for Fractional-Order Multi-Agent Systems without Intergroup Balance Condition. Fractal Fract. 2023, 7, 268. https://doi.org/10.3390/fractalfract7030268.
- Wei, Y.; Zhao, L.; Zhao, X.; Cao, J. Enhancing the Mathematical Theory of Nabla Tempered Fractional Calculus: Several Useful Equations. Fractal Fract. 2023, 7, 330. https://doi.org/10.3390/fractalfract7040330.
- Wang, M.; Wang, Y.; Chu, R. Dynamical Analysis of the Incommensurate Fractional-Order Hopfield Neural Network System and Its Digital Circuit Realization. Fractal Fract. 2023, 7, 474. https://doi.org/10.3390/fractalfract7060474.
References
- Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus: Models and Numerical Methods, 1st ed.; World Scientific: Singapore, 2012; ISBN 978-981-4355-20-9. [Google Scholar]
- Magin, R. Fractional Calculus in Bioengineering, 1st ed.; Begell House: Redding, CA, USA, 2006; ISBN 978-1567002157. [Google Scholar]
- Petráš, I. Fractional-Order Nonlinear Systems, 1st ed.; Springer: Heidelberg/Dordrecht, Germany; London, UK; New York, NY, USA, 2011; ISBN 978-3-642-18101-6. [Google Scholar]
- Sandev, T.; Tomovski, Z. Fractional Equations and Models. Theory and Applications, 1st ed.; Springer: Cham, Switzerland, 2019; ISBN 978-3-030-29616-2. [Google Scholar]
- Tarasov, V.E. Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, 1st ed.; Springer: Beijing, China, 2015; ISBN 978-3-642-14003-7. [Google Scholar]
- Almeida, R.; Tavares, D.; Torres, D.F.M. The Variable-Order Fractional Calculus of Variations, 1st ed.; Springer: Cham, Switzerland, 2019; ISBN 978-3-319-94005-2. [Google Scholar]
- Li, X.; Song, S. Impulsive Systems with Delays: Stability and Control, 1st ed.; Science Press & Springer: Singapore, 2022; ISBN 978-981-16-4686-7. [Google Scholar]
- Ostalczyk, P. Discrete Fractional Calculus: Applications in Control and Image Processing, 1st ed.; World Scientific: Singapore, 2015; ISBN 978-9814725668. [Google Scholar]
- Stamova, I.M.; Stamov, G.T. Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, 1st ed.; Taylor & Francis Group: Boca Raton, FL, USA, 2017; ISBN 9781498764834. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
MDPI and ACS Style
Stamova, I.; Stamov, G.; Li, X. Advances in Fractional-Order Neural Networks, Volume II. Fractal Fract. 2023, 7, 845. https://doi.org/10.3390/fractalfract7120845
AMA Style
Stamova I, Stamov G, Li X. Advances in Fractional-Order Neural Networks, Volume II. Fractal and Fractional. 2023; 7(12):845. https://doi.org/10.3390/fractalfract7120845
Chicago/Turabian StyleStamova, Ivanka, Gani Stamov, and Xiaodi Li. 2023. "Advances in Fractional-Order Neural Networks, Volume II" Fractal and Fractional 7, no. 12: 845. https://doi.org/10.3390/fractalfract7120845
