Enhanced evolutionary approach for solving fractional difference recurrent neural network systems: A comprehensive review and state of the art in view of time-scale analysis

Hanan S. Gafel; Saima Rashid; Hanan S. Gafel; Saima Rashid

doi:10.3934/math.20231571

AIMS Mathematics

2023, Volume 8, Issue 12: 30731-30759. doi: 10.3934/math.20231571

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Research article

Enhanced evolutionary approach for solving fractional difference recurrent neural network systems: A comprehensive review and state of the art in view of time-scale analysis

Hanan S. Gafel ¹,
Saima Rashid ^{2,3
,
,}

1.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
2.
Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
3.
Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, Lebanon

Received: 03 August 2023 Revised: 17 October 2023 Accepted: 27 October 2023 Published: 14 November 2023
MSC : 46S40, 47H10, 54H25

The present research deals with a novel three-dimensional fractional difference neural network model within undamped oscillations. Both the frequency and the amplitude of movements in equilibrium are subsequently estimated mathematically for such structures. According to the stability assessment, the thresholds of the fractional order were determined where bifurcations happen, and an assortment of fluctuations bifurcate within an insignificant equilibrium state. For such discrete fractional-order connections, the parameterized spectrum of undamped resonances is also predicted, and the periodicity and strength of variations are calculated computationally and numerically. Several qualitative techniques, including the Lyapunov exponent, phase depictions, bifurcation illustrations, the $ 0-1 $ analysis and the approximate entropy technique, have been presented with the rigorous analysis. These outcomes indicate that the suggested discrete fractional neural network model has crucial as well as complicated dynamic features that have been affected by the model's variability, both in commensurate and incommensurate cases. Furthermore, the approximation entropy verification and $ \mathbb{C}_{0} $ procedure are used to assess variability and confirm the emergence of chaos. Ultimately, irregular controllers for preserving and synchronizing the suggested framework are highlighted.
- discrete fractional difference equations,
- undamped oscillations,
- bifurcation,
- chaos control,
- master and slave system,
- time scale
Citation: Hanan S. Gafel, Saima Rashid. Enhanced evolutionary approach for solving fractional difference recurrent neural network systems: A comprehensive review and state of the art in view of time-scale analysis[J]. AIMS Mathematics, 2023, 8(12): 30731-30759. doi: 10.3934/math.20231571

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Abstract

The present research deals with a novel three-dimensional fractional difference neural network model within undamped oscillations. Both the frequency and the amplitude of movements in equilibrium are subsequently estimated mathematically for such structures. According to the stability assessment, the thresholds of the fractional order were determined where bifurcations happen, and an assortment of fluctuations bifurcate within an insignificant equilibrium state. For such discrete fractional-order connections, the parameterized spectrum of undamped resonances is also predicted, and the periodicity and strength of variations are calculated computationally and numerically. Several qualitative techniques, including the Lyapunov exponent, phase depictions, bifurcation illustrations, the $ 0-1 $ analysis and the approximate entropy technique, have been presented with the rigorous analysis. These outcomes indicate that the suggested discrete fractional neural network model has crucial as well as complicated dynamic features that have been affected by the model's variability, both in commensurate and incommensurate cases. Furthermore, the approximation entropy verification and $ \mathbb{C}_{0} $ procedure are used to assess variability and confirm the emergence of chaos. Ultimately, irregular controllers for preserving and synchronizing the suggested framework are highlighted.

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