Fractal and Fractional

23 pages, 3533 KB

Open AccessArticle

Nabla Fractional Distributed Nash Seeking for Non-Cooperative Games

by Yao Xiao, Sunming Ge, Yihao Qiao, Tieqiang Gang and Lijie Chen

Fractal Fract. 2025, 9(12), 756; https://doi.org/10.3390/fractalfract9120756 - 21 Nov 2025

This paper pioneers the introduction of nabla fractional calculus into distributed Nash equilibrium (NE) seeking for non-cooperative games (NGs), proposing several novel discrete-time fractional-order algorithms. We first develop a gradient play-based algorithm under perfect information and subsequently extend it to partial-information settings. Two types of communication network topologies among agents, namely connected undirected graphs and strongly connected unbalanced directed graphs, are explicitly considered. When the pseudo-gradient mapping of the NG is Lipschitz continuous and strongly monotone, the proposed algorithms are proven to achieve asymptotic convergence to the NE with at least a Mittag–Leffler convergence rate. Both the step size and the fractional order act as tunable parameters that jointly influence the convergence performance. Numerical experiments on potential games and Nash–Cournot games demonstrate the effectiveness of the proposed algorithms. Full article

(This article belongs to the Special Issue Distributed Control and Optimization of Fractional-Order Systems: Theory and Applications)

34 pages, 8174 KB

Open AccessArticle

Formation Control of Underactuated AUVs Based on Event-Triggered Communication and Fractional-Order Sliding Mode Control

by Long He, Ya Zhang, Shizhong Li, Bo Li, Mengting Xie, Zehui Yuan and Chenrui Bai

Fractal Fract. 2025, 9(12), 755; https://doi.org/10.3390/fractalfract9120755 - 21 Nov 2025

Abstract

To address the challenges faced by multiple autonomous underwater vehicles (AUVs) in formation control under complex marine environments—such as model uncertainties, external disturbances, dynamic communication topology variations, and limited communication resources—this paper proposes an integrated control framework that combines robust individual control, distributed cooperative formation, and dynamic event-triggered communication. At the individual control level, a robust control method based on a fractional-order sliding mode observer (FOSMO) and a fractional-order terminal sliding mode controller (FOTSMC) is developed. The observer exploits the memory and broadband characteristics of fractional calculus to achieve high-precision estimation of lumped disturbances, while the controller constructs a non-integer-order sliding surface with an adaptive gain law to guarantee finite-time convergence of tracking errors. At the formation coordination level, a distributed trajectory generation method based on dynamic consensus is proposed to achieve reference trajectory planning and formation maintenance in a cooperative manner. At the communication level, a dynamic-threshold event-triggered mechanism is designed, where the triggering condition is adaptively adjusted according to the state errors, thereby significantly reducing communication load and energy consumption. Theoretically, Lyapunov-based analysis rigorously proves the stability and convergence of the closed-loop system. Numerical simulations confirm that the proposed method outperforms several benchmark algorithms in terms of tracking accuracy and disturbance rejection. Moreover, the integrated framework maintains precise formation under communication topology variations, achieving a communication reduction rate exceeding 65% compared to periodic protocols while preserving coordination accuracy. Full article

(This article belongs to the Special Issue Fractional Control Systems and Estimation: Control, State Observers and Parameter Estimation)

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27 pages, 630 KB

Open AccessArticle

Fractional Modeling and Stability Analysis of Tomato Yellow Leaf Curl Virus Disease: Insights for Sustainable Crop Protection

by Mansoor Alsulami, Ali Raza, Marek Lampart, Umar Shafique and Eman Ghareeb Rezk

Fractal Fract. 2025, 9(12), 754; https://doi.org/10.3390/fractalfract9120754 - 21 Nov 2025

Abstract

Tomato Yellow Leaf Curl Virus (TYLCV) has recently caused severe economic losses in global tomato production. According to the International Plant Protection Convention (IPPC), yield reductions of 50–60% have been reported in several regions, including the Caribbean, Central America, and South Asia, with losses in sensitive cultivars reaching up to 90–100%. In developing countries, TYLCV and mixed infections affect more than seven million hectares of tomato-growing land annually. In this study, we construct and analyze a nonlinear dynamic model describing the transmission of TYLCV, incorporating the Caputo fractional-order derivative operator. The existence and uniqueness of solutions to the proposed model are rigorously established. Equilibrium points are identified, and the Jacobian determinant approach is applied to compute the basic reproduction number,

R_{0}

. Suitable Lyapunov functions are formulated to analyze the global asymptotic stability of both the disease-free and endemic equilibria. The model is numerically solved using the Grünwald–Letnikov-based nonstandard finite difference method, and simulations assess how the memory index and preventive strategies influence disease propagation. The results reveal critical factors governing TYLCV transmission and suggest effective intervention measures to guide sustainable crop protection policies. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

21 pages, 1007 KB

Open AccessArticle

Ulam-Type Stability and Krasnosel’skii’s Fixed Point Approach for φ-Caputo Fractional Neutral Differential Equations with Iterated State-Dependent Delays

by Ravi P. Agarwal, Mihail M. Konstantinov and Ekaterina B. Madamlieva

Fractal Fract. 2025, 9(12), 753; https://doi.org/10.3390/fractalfract9120753 - 21 Nov 2025

Abstract

This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order

α \in (0, 1)

with respect to a strictly increasing function

φ

, [...] Read more.

α \in (0, 1)

with respect to a strictly increasing function

φ

, the analysis extends classical results to nonuniform memory. The neutral term and delay chain are defined recursively by the solution, with arbitrary continuous initial data. Existence and uniqueness of solutions are established using Krasnosel’skii’s fixed point theorem. Sufficient conditions for Ulam–Hyers stability are obtained via the Volterra-type integral form and a

φ

-fractional Grönwall inequality. Examples illustrate both standard and nonlinear time scales, including a Hopfield neural network with iterated delays, which has not been previously studied even for integer-order equations. Fractional neural networks with iterated state-dependent delays provide a new and effective model for the description of AI processes—particularly machine learning and pattern recognition—as well as for modelling the functioning of the human brain. Full article

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Fractal Fract., Volume 9, Issue 12 (December 2025) – 4 articles

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