The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system's solutions, we investigate stochasticity's influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

Computer Engineering Department College of Computer and Information Sciences King Saud University Riyadh Saudi Arabia

Department of Computer Science and Mathematics Lebanese American University Byblos Lebanon

Department of Mathematics City University of Science and Information Technology Peshawar KP 2500 Pakistan

IT4Innovations VSB Technical University of Ostrava Ostrava Czech Republic

School of Mathematics and Data Sciences Changji University Changji 831100 Xinjiang People's Republic of China

Software Engineering Department College of Computer and Information Sciences King Saud University Riyadh Saudi Arabia

Zobrazit více v PubMed

Daftardar-Gejji V, editor. Fractional Calculus. Alpha Science International Limited; 2013.

Goodrich C, Peterson AC. Discrete Fractional Calculus. Springer; 2015.

David SA, Linares JL, de Jesus Agnolon Pallone EM. Fractional order calculus: Historical apologia, basic concepts and some applications. Revista Brasileira de Ensino de Física. 2011;33:4302. doi: 10.1590/S1806-11172011000400002. DOI

Almeida R, Tavares D, Torres DFM. The Variable-Order Fractional Calculus of Variations. Springer; 2019.

Bagley RL, Torvik PJ. Fractional calculus—A different approach to the analysis of viscoelastically damped structures. AIAA J. 1983;21(5):741–748. doi: 10.2514/3.8142. DOI

West BJ. Colloquium: Fractional calculus view of complexity: A tutorial. Rev. Mod. Phys. 2014;86(4):1169. doi: 10.1103/RevModPhys.86.1169. DOI

Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Thir Solution and Some of Thir Applications. Elsevier; 1998.

Alexander, S., Cunningham, W. J., Lanier, J., Smolin, L., Stanojevic, S., Toomey, M. W., & Wecker, D. (2021). The autodidactic universe. arXiv preprint arXiv:2104.03902.

Lewandowski R, Litewka P, Lasecka-Plura M, Pawlak ZM. Dynamics of structures, frames, and plates with viscoelastic dampers or layers: A literature review. Buildings. 2023;13(9):2223. doi: 10.3390/buildings13092223. DOI

Haug, J. C. Towards Reliable Machine Learning in Evolving Data Streams. Diss. Universität Tübingen (2022).

Park JH, Lee TH, Liu Y, Chen J. Dynamic Systems with Time Delays: Stability and Control. Springer; 2019.

Rajchakit G, Agarwal P, Ramalingam S. Stability Analysis of Neural Networks. Springer; 2021.

Lu Q. Relationship between the nonlinear oscillator and the motor cortex. IEEE Access. 2019;7:44525–44535. doi: 10.1109/ACCESS.2019.2908719. DOI

Lu Q. Dynamics and coupling of fractional-order models of the motor cortex and central pattern generators. J. Neural Eng. 2020;17(3):036021. doi: 10.1088/1741-2552/ab8dd6. PubMed DOI

Spielmans, G. I. & Ellefson, E. M. S. Effects, questionable outcomes: Bremelanotide for hypoactive sexual desire disorder. J. Sex Res. 1–22 (2023). PubMed

Thelen, E. & Smith, L. B. Dynamic systems theories. In Handbook of Child Psychology, Vol. 1 (2007).

Solmaz B, Moore BE, Shah M. Identifying behaviors in crowd scenes using stability analysis for dynamical systems. IEEE Trans. Pattern Anal. Mach. Intell. 2012;34(10):2064–2070. doi: 10.1109/TPAMI.2012.123. PubMed DOI

Gauthier Marquis, J.-C. Fragmentation et déformation de la banquise. Diss. Université du Québec à Rimouski (2018).

Tarasov VE. On history of mathematical economics: Application of fractional calculus. Mathematics. 2019;7(6):509. doi: 10.3390/math7060509. DOI

Bocharov GA, Rihan FA. Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 2000;125(1–2):183–199. doi: 10.1016/S0377-0427(00)00468-4. DOI

Iqbal SA, Hafez MG, Chu YM, Park C. Dynamical analysis of nonautonomous RLC circuit with the absence and presence of Atangana–Baleanu fractional derivative. J. Appl. Anal. Comput. 2022;12(2):770–789.

Atangana A, Francisco Gómez-Aguilar J. Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus. 2018;133:1–22. doi: 10.1140/epjp/i2018-12021-3. DOI

Williams RA. Lessons learned on development and application of agent-based models of complex dynamical systems. Simul. Model. Pract. Theory. 2018;83:201–212. doi: 10.1016/j.simpat.2017.11.001. DOI

Namdari A, Li Z. A review of entropy measures for uncertainty quantification of stochastic processes. Adv. Mech. Eng. 2019;11(6):1687814019857350. doi: 10.1177/1687814019857350. DOI

Shojaie A, Fox EB. Granger causality: A review and recent advances. Annu. Rev. Stat. Appl. 2022;9:289–319. doi: 10.1146/annurev-statistics-040120-010930. PubMed DOI PMC

Yuan Q. Fast Kinetic Studies of Electron Transfer Reactions for Cytochrome BC1 and Cytochrome B5. University of Arkansas; 2009.

Khan, S. U. & Ali, I. Numerical analysis of stochastic SIR model by Legendre spectral collocation method. In Advances in Mechanical Engineering, Vol. 11, 7 (SAGE Publications Sage UK, 2019).

Ali I, Khan SU. Analysis of stochastic delayed SIRS model with exponential birth and saturated incidence rate. Chaos Solitons Fractals. 2020;138:110008. doi: 10.1016/j.chaos.2020.110008. DOI

Khan SU, Ali I. Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind. Comput. Appl. Math. 2019;38(3):125. doi: 10.1007/s40314-019-0897-2. DOI

Khan SU, Ali I. Applications of Legendre spectral collocation method for solving system of time delay differential equations. Adv. Mech. Eng. 2020;12(6):1687814020922113. doi: 10.1177/1687814020922113. DOI

Algehyne EA, Khan FU, Khan SU, Jamshed W, Tag El Din ESM. Dynamics of stochastic Zika virus with treatment class in human population via spectral method. Symmetry. 2022;14(10):2137. doi: 10.3390/sym14102137. DOI

Ali I, Khan SU. A dynamic competition analysis of stochastic fractional differential equation arising in finance via pseudospectral method. Mathematics. 2023;11(6):1328. doi: 10.3390/math11061328. DOI

Das P, Das P, Mukherjee S. Stochastic dynamics of Michaelis–Menten kinetics based tumor-immune interactions. Physica A Stat. Mech. Appl. 2020;541:123603. doi: 10.1016/j.physa.2019.123603. DOI

Das P, Upadhyay RK, Das P, Ghosh D. Exploring dynamical complexity in a time-delayed tumor-immune model. Chaos Interdiscip. J. Nonlinear Sci. 2020;30(12):123118. doi: 10.1063/5.0025510. PubMed DOI

He L, Banihashemi S, Jafari H, Babaei A. Numerical treatment of a fractional order system of nonlinear stochastic delay differential equations using a computational scheme. Chaos Solitons Fractals. 2021;149:111018. doi: 10.1016/j.chaos.2021.111018. DOI

Banihashemi, S., Jafari, H. & Babaei, A. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Disc. Continue. Dyn. Syst. S10 (2021).

Xue F, et al. Integrated memory devices based on 2D materials. Adv. Mater. 2022;34(48):2201880. doi: 10.1002/adma.202201880. PubMed DOI