Global Mittag–Leffler stability of coupled system of fractional-order differential equations on network

Liu, Hongli; Jiang, Yao‐Lin; Wang, Zuolei; Zhang, Long; Teng, Zhidong

doi:10.1016/j.amc.2015.08.043

Cited by 48 publications

(26 citation statements)

References 28 publications

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“…From Lemma 3.1, we obtain lim t→∞ U(t) = 0; obviously, lim t→∞ e(t) = 0, that is, system (12) can achieve synchronization under adaptive law (11). The proof is completed.…”

Section: C(t) Pĝ +ĝmentioning

confidence: 60%

“…These research efforts have shown that, compared with integer calculus, fractional calculus has a greater advantage in describing the memory and hereditary properties of manifold material and processes, and fractional calculus has plenty of freedom when we simulate real-world problems. In recent years, there has been a great deal of work to study fractional-order systems in dynamics and control [9][10][11]. In the real world, the complex networks are composed of a large number of interconnected fractional-order dynamical units; therefore, it is necessary to investigate fractional-order complex dynamical networks.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pinning and adaptive synchronization of fractional-order complex dynamical networks with and without time-varying delay

Wang

Ruan

et al. 2018

Adv Differ Equ

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The synchronization problem for a class of fractional-order complex dynamical networks with and without time-varying delay is investigated in this paper. By utilizing generalized Barbalat's lemma, Razumikhin-type stability theory and matrix inequality technique, some sufficient criteria ensuring synchronization under pinning control and pinning adaptive feedback control are derived. Finally, three numerical simulations are presented to demonstrate the effectiveness of the obtained results.

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“…From Lemma 3.1, we obtain lim t→∞ U(t) = 0; obviously, lim t→∞ e(t) = 0, that is, system (12) can achieve synchronization under adaptive law (11). The proof is completed.…”

Section: C(t) Pĝ +ĝmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Pinning and adaptive synchronization of fractional-order complex dynamical networks with and without time-varying delay

Wang

Ruan

et al. 2018

Adv Differ Equ

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show abstract

“…This result is different from the previous studies. Firstly, Theorem 3.1 is a generalization of the main result of Li [7]. The model in this paper is more complicated for every vertex and the conditions of Theorem 3.1 are different from the result in Li [7].…”

Section: Conclusion and Utlooksmentioning

confidence: 99%

“…In fact, it is more valuable and practical to investigate a coupled system of fractional-order differential equations on the network. Recently, Li [7] investigated the global Mittag-Leffler stability of the following coupled system of fractional-order differential equations on network (CSFDEN):…”

Section: Introductionmentioning

confidence: 99%

Mittag–Leffler stability for a new coupled system of fractional-order differential equations on network

Gao

2018

Adv Differ Equ

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In this paper, the stability problem of a new coupled model constructed by two fractional-order differential equations for every vertex is studied. The coupled relationship is hybrid. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, sufficient conditions that the coexistence equilibrium of the coupling model is globally Mittag-Leffler stable in R 2n are derived. An example is given to illustrate the main results.

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“…Because, non-integer order differential equations have many applications in various branches of science and engineering such as signal processing, viscoelasticity, biology, physics, chemistry, control theory and stability of networking and modeling of biological phenomenons, etc., for detail see [6,13,14,16,18]. The qualitative theory devoted to existence of solutions to non-integer order differential equations involving boundary conditions has been an active area of research for the last few decades.…”

Section: Introductionmentioning

confidence: 99%

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

Ali¹,

Shah²,

Khan³

2017

J. Nonlinear Sci. Appl.

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This paper is concerned with developing some conditions that reveal existing and stability analysis for solutions to a class of differential equations with fractional order. The required conditions are obtained by applying the technique of degree theory of topological type. The concerned problem is converted to the integral equation and then to operator equation, where the operator is defined by T : C[0, 1] → C[0, 1]. It should be noted that the assumptions on nonlinear function f(t, u(t)) does not usually ascertain that the operator T being compact. Moreover, in this paper we also establish some conditions under which the solution of the considered class is Hyers-Ulam stable and also satisfies the conditions of Hyers-Ulam-Rassias and generalized Hyers-Ulam stability. Proper example is provided for the illustration of main results.

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Global Mittag–Leffler stability of coupled system of fractional-order differential equations on network

Cited by 48 publications

References 28 publications

Pinning and adaptive synchronization of fractional-order complex dynamical networks with and without time-varying delay

Pinning and adaptive synchronization of fractional-order complex dynamical networks with and without time-varying delay

Mittag–Leffler stability for a new coupled system of fractional-order differential equations on network

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

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