Analyzing finite-time convergence for variable-order fractional discrete dynamics in Degn–Harrison reaction–diffusion systems

Shaher Momani; Iqbal H. Jebril; Iqbal M. Batiha; Lina S. Calucag; Anjan Biswas; Shaher Momani; Iqbal H. Jebril; Iqbal M. Batiha; Lina S. Calucag; Anjan Biswas

doi:10.3934/math.2026501

AIMS Mathematics

2026, Volume 11, Issue 4: 12204-12232. doi: 10.3934/math.2026501

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Analyzing finite-time convergence for variable-order fractional discrete dynamics in Degn–Harrison reaction–diffusion systems

1.
Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan
2.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates
3.
Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan
4.
Department of Mathematics and Science, University of Technology Bahrain, Salmabad 18041, Bahrain
6.
Department of Mathematics & Physics, Grambling State University, Grambling, LA 71245–2715, USA
7.
Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon–61080, Türkiye
8.
Department of Physics and Electronics, Khazar University, Baku, AZ–1096, Azerbaijan
9.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa–0204, Pretoria, South Africa

Received: 24 January 2026 Revised: 26 March 2026 Accepted: 10 April 2026 Published: 30 April 2026

This study presents a novel investigation into finite-time stability (FTS) and synchronization phenomena within a discrete reaction–diffusion system (RDS) governed by variable-order fractional (VOF) operators, inspired by the Degn–Harrison (D–H) model. By employing Caputo-type VOF differences, we model memory effects and time-varying dynamics typical of complex biological and chemical processes. Theoretical contributions include rigorous Lyapunov function (LF)-based criteria for establishing tempered Mittag-Leffler stability (MLS) and global FTS, as well as explicit expressions for the settling time $ T^{*} $. A fractional-order (FO) error system is also analyzed, demonstrating that linear coupling ensures finite-time synchronization under variable-order conditions. Extensive numerical simulations confirm the theoretical predictions across various FO profiles $ \delta(t) $ and parameter regimes. These findings bridge discrete fractional modeling with practical control strategies for systems exhibiting hereditary and anomalous diffusion effects.
- finite-time stability,
- variable-order fractional operators,
- reaction-diffusion systems,
- Lyapunov methods,
- discrete Caputo difference,
- Mittag-Leffler stability
Citation: Shaher Momani, Iqbal H. Jebril, Iqbal M. Batiha, Lina S. Calucag, Anjan Biswas. Analyzing finite-time convergence for variable-order fractional discrete dynamics in Degn–Harrison reaction–diffusion systems[J]. AIMS Mathematics, 2026, 11(4): 12204-12232. doi: 10.3934/math.2026501

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Abstract

This study presents a novel investigation into finite-time stability (FTS) and synchronization phenomena within a discrete reaction–diffusion system (RDS) governed by variable-order fractional (VOF) operators, inspired by the Degn–Harrison (D–H) model. By employing Caputo-type VOF differences, we model memory effects and time-varying dynamics typical of complex biological and chemical processes. Theoretical contributions include rigorous Lyapunov function (LF)-based criteria for establishing tempered Mittag-Leffler stability (MLS) and global FTS, as well as explicit expressions for the settling time $ T^{*} $. A fractional-order (FO) error system is also analyzed, demonstrating that linear coupling ensures finite-time synchronization under variable-order conditions. Extensive numerical simulations confirm the theoretical predictions across various FO profiles $ \delta(t) $ and parameter regimes. These findings bridge discrete fractional modeling with practical control strategies for systems exhibiting hereditary and anomalous diffusion effects.

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