Asymptotic stability of distributed order nonlinear dynamical systems

Fernández‐Anaya, Guillermo; Nava-Antonio, Guillermo; Jamous-Galante, J.; Muñoz-Vega, Rodrígo; Hernández‐Martínez, Eduardo Gamaliel

doi:10.1016/j.cnsns.2017.01.020

Cited by 47 publications

(40 citation statements)

References 20 publications

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“…The Lyapunov direct method, used for analysis of stability, was first generalized for nonlinear time-varying DO systems in [ 355 , 356 , 357 ] and was used to determine the stability or asymptotic stability of certain nonlinear systems including a DO analog of the Lorenz system. The theoretical framework proposed in the studies [ 355 , 356 ] was then used to analyze different nonlinear time-varying DO systems including a DO consensus model [ 358 ], the DO Lorenz system [ 359 ], and the DO Van der Pol oscillator [ 330 , 360 ]. The consensus of multi-agent systems with fixed directed graphs and described by DODE, was analyzed in [ 358 ] and sufficient conditions were obtained for robust consensus in the presence and absence of external disturbances.…”

Section: Applications To Control Theorymentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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Section: Applications To Control Theorymentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…e following section includes some definitions of the fractional order and the distributed order derivatives [16,22,33,34], with useful remark and theorem that will be used later.…”

Section: Preliminariesmentioning

confidence: 99%

“…By transforming system (16) by Laplace and applying Remark 1 in L e i (t) � E i (s), i � 1, 2, 3, 4, then we get:…”

Section: Chaos Synchronization Of System (10)mentioning

confidence: 99%

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The Dynamics Behavior of Coupled Generalized van der Pol Oscillator with Distributed Order

Themairi

Farghaly

2020

Mathematical Problems in Engineering

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In this paper, we presented different behaviors such as chaotic and hyperchaotic of the generalized van der Pol oscillator with distributed order. We introduced the parameter intervals of these behaviors by computing the Lyapunov exponents of the oscillator, which is a good test for classifying the dynamical systems’ solutions. The active control approach with the Laplace transform technique was used to realize the antisynchronization and control of the proposed oscillator. Finally, numerical investigations have been carried out on the dynamics of the proposed oscillator to verify the reliability of our analytical results.

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“…By using the properties of Mittag-Leffler function, Laplace transform, and the inequality analysis technique, the authors in previous studies 10,11 established several sufficient conditions for guaranteeing Mittag-Leffler stability of nonlinear Caputo and Riemann-Liouville fractional neutral singular systems, respectively. By applying the Lyapunov direct method, the authors in previous studies [12][13][14][15][16][17][18][19] obtained several sufficient conditions on the asymptotical stability and synchronization of Caputo and Riemann-Liouville fractional nonlinear systems without and with delays, respectively.…”

Section: Introductionmentioning

confidence: 99%

Asymptotical stability analysis of Riemann‐Liouville q‐fractional neutral systems with mixed delays

Yang

Alsaedi

Hayat

et al. 2019

Math Methods in App Sciences

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This paper investigates the asymptotical stability of Riemann‐Liouville q‐fractional neutral systems with mixed delays (constant time delay and distributed delay). By constructing some appropriate Lyapunov‐Kravsovskii functionals, some sufficient conditions on delay‐dependent and delay‐independent asymptotical stability are obtained in terms of linear matrix inequality (LMI). Our employed method is based on the direct calculation of quantum derivatives of the Lyapunov‐Kravsovskii functionals. Finally, two examples are presented to demonstrate the availability of our obtained results.

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Asymptotic stability of distributed order nonlinear dynamical systems

Cited by 47 publications

References 20 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

The Dynamics Behavior of Coupled Generalized van der Pol Oscillator with Distributed Order

Asymptotical stability analysis of Riemann‐Liouville q‐fractional neutral systems with mixed delays

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