Synchronization of Short-Memory Fractional Directed Higher-Order Networks
Abstract
1. Introduction
- A new complex network framework featuring directed, high-order interactions governed by short-memory fractional dynamics is introduced. Unlike traditional models [13,14,15], more diverse and realistic connection patterns are captured by this formulation, enabling deeper insights into the collective behavior of complex systems.
- The synchronization of a generally short-memory fractional directed higher-order network using a pinning control scheme is investigated.
- An improved predictor–corrector algorithm for solving short-memory fractional systems is proposed. Based on this algorithm, a numerical simulation is conducted to verify the effectiveness of the derived theoretical results.
2. Preliminaries
3. Main Result
3.1. Model Description
3.2. Synchronization of Short-Memory Directed Higher-Order Networks
3.3. Adaptive Synchronization of Short-Memory Fractional Higher-Order Directed Networks
4. Numerical Implementation
4.1. Predictor–Corrector Algorithm
4.2. Numerical Examples
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Battiston, F.; Amico, E.; Barrat, A.; Bianconi, G.; de Arruda, G.F.; Franceschiello, B.; Iacopini, I.; Kéfi, S.; Latora, V.; Moreno, Y.; et al. The physics of higher-order interactions in complex systems. Nat. Phys. 2021, 17, 1093–1098. [Google Scholar] [CrossRef]
- Cencetti, G.; Battiston, F.; Lepri, B.; Karsai, M. Temporal properties of higher-order interactions in social networks. Sci. Rep. 2021, 11, 7028. [Google Scholar] [CrossRef]
- Huang, X.; Xu, K.; Chu, C.; Jiang, T.; Yu, S. Weak higher-order interactions in macroscopic functional networks of the resting brain. J. Neurosci. 2017, 37, 10481–10497. [Google Scholar] [CrossRef] [PubMed]
- Li, Z.; Ma, W.; Ma, N. Partial topology identification of tempered fractional-order complex networks via synchronization method. Math. Methods Appl. Sci. 2023, 46, 3066–3079. [Google Scholar] [CrossRef]
- Bick, C.; Gross, E.; Harrington, H.A.; Schaub, M.T. What are higher-order networks? SIAM Rev. 2023, 65, 686–731. [Google Scholar] [CrossRef]
- Battiston, F.; Cencetti, G.; Iacopini, I.; Latora, V.; Lucas, M.; Patania, A.; Young, J.-G.; Petri, G. Networks beyond pairwise interactions: Structure and dynamics. Phys. Rep. 2020, 874, 1–92. [Google Scholar] [CrossRef]
- Majhi, S.; Perc, M.; Ghosh, D. Dynamics on higher-order networks: A review. J. R. Soc. Interface 2022, 19, 20220043. [Google Scholar] [CrossRef]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scince: Singapore, 2000. [Google Scholar] [CrossRef]
- Wei, Y.; Chen, Y.; Cheng, S.; Wang, Y. A note on short memory principle of fractional calculus. Fract. Calc. Appl. Anal. 2017, 20, 1382–1404. [Google Scholar] [CrossRef]
- Lakshmikantham, V. Theory of Integro-Differential Equations; CRC Press: Boca Raton, FL, USA, 1995. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Shiri, B.; Wu, G.; Baleanu, D. Applications of short memory fractional differential equations with impulses. Discontinuity Nonlinearity Complex. 2023, 12, 167–182. [Google Scholar] [CrossRef]
- Gu, C.; Zheng, F.; Shiri, B. Mittag-Leffler stability analysis of tempered fractional neural networks with short memory and variable-order. Fractals 2021, 29, 2140029. [Google Scholar] [CrossRef]
- Wu, G.; Luo, M.; Huang, L.; Banerjee, S. Short memory fractional differential equations for new memristor and neural network design. Nonlinear Dyn. 2020, 100, 3611–3623. [Google Scholar] [CrossRef]
- Diethelm, K.; Ford, N.J.; Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002, 29, 3–22. [Google Scholar] [CrossRef]
- Wu, G.; Deng, Z.; Baleanu, D.; Zeng, D.Q. New variable-order fractional chaotic systems for fast image encryption. Chaos 2019, 29, 083103. [Google Scholar] [CrossRef] [PubMed]
- Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D. Complex networks: Structure and dynamics. Phys. Rep. 2006, 424, 175–308. [Google Scholar] [CrossRef]
- Chen, J.; Sun, W.; Zheng, S. New predefined-time stability theorem and synchronization of fractional-order memristive delayed BAM neural networks. Commun. Nonlinear Sci. Numer. Simul. 2025, 148, 108850. [Google Scholar] [CrossRef]
- Arenas, A.; Díaz-Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C. Synchronization in complex networks. Phys. Rep. 2008, 469, 93–153. [Google Scholar] [CrossRef]
- Skardal, P.S.; Arenas, A. Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Commun. Phys. 2020, 3, 218. [Google Scholar] [CrossRef]
- Saçu, İ.E. Effects of high-order interactions on synchronization of a fractional-order neural system. Cogn. Neurodyn. 2024, 18, 1877–1893. [Google Scholar] [CrossRef]
- Ramasamy, M.; Kumarasamy, S.; Srinivasan, A.; Subburam, P.; Rajagopal, K. Dynamical effects of hypergraph links in a network of fractional-order complex systems. Chaos 2022, 2, 123128. [Google Scholar] [CrossRef]
- Tatar, N.E. Stability and synchronization of a fractional neutral higher-order neural network system. Int. J. Nonlinear Sci. Numer. Simul. 2020, 21, 443–458. [Google Scholar] [CrossRef]
- Ma, C.; Ma, W.; Wang, X. Synchronization on fractional multiplex higher-order networks. Chaos 2024, 34, 103134. [Google Scholar] [CrossRef] [PubMed]
- Hai, X.; Yu, Y.; Xu, C.; Ren, G. Stability analysis of fractional differential equations with the short-term memory property. Fract. Calc. Appl. Anal. 2022, 25, 962–994. [Google Scholar] [CrossRef]
- DeLellis, P.; di Bernardo, M.; Russo, G. On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Trans. Circuits Syst. 2010, 58, 576–583. [Google Scholar] [CrossRef]
- Lu, W.; Chen, T. QUAD-condition, synchronization, consensus of multiagents, and anti-synchronization of complex networks. IEEE Trans. Cybern. 2019, 1, 3384–3388. [Google Scholar] [CrossRef] [PubMed]
- Lu, W.; Chen, T. New approach to synchronization analysis of linearly coupled ordinary differential systems. Phys. D 2006, 213, 214–230. [Google Scholar] [CrossRef]
- Golub, G.H.; Van Loan, C.F. Matrix Computations; JHU Press: Baltimore, ML, USA, 2013. [Google Scholar]
- Wang, X.; Ding, X.; Li, J.; Cao, J. Multisynchronization of delayed fractional-order neural networks via average impulsive interval. Neural Process. Lett. 2023, 55, 12437–12457. [Google Scholar] [CrossRef]
- Yu, W.; Chen, G.; Lü, J. On pinning synchronization of complex dynamical networks. Automatica 2009, 45, 429–435. [Google Scholar] [CrossRef]
- Diethelm, K.; Ford, N.J.; Freed, A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms 2004, 36, 31–52. [Google Scholar] [CrossRef]
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. |
© 2025 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
Wang, X.; Ma, W.; Zou, J. Synchronization of Short-Memory Fractional Directed Higher-Order Networks. Fractal Fract. 2025, 9, 440. https://doi.org/10.3390/fractalfract9070440
Wang X, Ma W, Zou J. Synchronization of Short-Memory Fractional Directed Higher-Order Networks. Fractal and Fractional. 2025; 9(7):440. https://doi.org/10.3390/fractalfract9070440
Chicago/Turabian StyleWang, Xiaoqin, Weiyuan Ma, and Jiayu Zou. 2025. "Synchronization of Short-Memory Fractional Directed Higher-Order Networks" Fractal and Fractional 9, no. 7: 440. https://doi.org/10.3390/fractalfract9070440
APA StyleWang, X., Ma, W., & Zou, J. (2025). Synchronization of Short-Memory Fractional Directed Higher-Order Networks. Fractal and Fractional, 9(7), 440. https://doi.org/10.3390/fractalfract9070440