This paper studies the synchronization of fractional-order neural networks with multiple time delays. Based on an inequality of fractional-order and comparison principles of linear fractional equation with multiple time delays, some sufficient conditions for synchronization of master-slave systems are obtained. Example and related simulations are given to demonstrate the feasibility of the theoretical results

Alsaedi, Ahmed

Cao, Jinde

Hayat, Tasawar

Wu, Ranchao

Zhang, Weiwei

English

Nonlinear Analysis: Modelling and Control

ISSN 1392-5113https://doi.org/10.15388/NA.2017.5.4Nonlinear Analysis: Modelling and Control, 2017, Vol. 22, No. 5, 636–645Synchronization of a class of fractional-order neuralnetworks with multiple time delays by comparisonprinciples∗Weiwei Zhanga, Ranchao Wub, Jinde Caoc,d,1, Ahmed Alsaedid,Tasawar Hayatd,eaSchool of Mathematics and Computational Science,Anqing Normal University,AnQing 246011, Chinawwzhahu@aliyun.combSchool of Mathematics, Anhui University,Hefei 230039, Chinarcwu@ahu.edu.cncSchool of Mathematics, Southeast University,Nanjing 210096, Chinajdcao@seu.edu.cndDepartment of Mathematics, Faculty of Science,King Abdulaziz University,Jeddah 21589, Saudi Arabiaaalsaedi@hotmail.comeDepartment of Mathematics, Quaid-I-Azam University,Islamabad 44000, Pakistantahaksag@yahoo.comReceived: Octobre 8, 2016 / Revised: April 23, 2017 / Published online: September 24, 2017Abstract. This paper studies the synchronization of fractional-order neural networks with multipletime delays. Based on an inequality of fractional-order and comparison principles of linear frac-tional equation with multiple time delays, some sufficient conditions for synchronization of master-slave systems are obtained. Example and related simulations are given to demonstrate the feasibilityof the theoretical results.Keywords: fractional-order neural networks, comparison principles, multiple time delays,synchronization.∗This research was supported by the National Natural Science Foundation of China under grantNo. 11571016, the Natural Science Foundation of Anhui Province under grant No. 1608085MA14, andthe Natural Science Foundation of the Higher Education Institutions of Anhui Province under grantNo. KJ2015A152.1Corresponding author.c© Vilnius University, 2017Synchronization of a class of fractional-order neural networks 6371 IntroductionNeural networks are important nonlinear systems, which have been studied in scientificresearch and have been applied to the signal processing, parallel computation, optimiza-tion, artificial intelligence, and so on. It is known that fractional calculus mainly dealswith a generalization of differentiation and integration of arbitrary orders. Comparedwith classical integer-order models, the fractional-order calculus owns the description ofmemory and hereditary properties of a variety of processes. Fractional-order models arefar better regard the dynamical behaviors of systems [2,6,13,15,16,23,24]. Taking thesefactors into account, many researchers have incorporated fractional calculus to neuralnetworks and formed fractional-order neural networks [3, 7–9, 14, 25, 27].Time-delayed models unavoidable exist in biological, engineering systems, and neuralnetworks, see [19–21, 28, 29]. Generally, time delays will affect oscillation and instabil-ity behavior of a network. Nowadays, the synchronization of fractional-order delayedneural networks has attracted more and more attention. Some synchronization resultshave been obtained, for instance, in [26], authors studied the finite-time synchroniza-tion of fractional-order memristor-based neural networks with time delay. Paper [10]investigated the stability and synchronization of memristor-based fractional-order delayedneural networks. In [4], the synchronization of fractional-order complex-valued neuralnetworks with time delay was studied. Paper [5] discussed the adaptive synchronizationof fractional-order memristor-based neural networks with time delay.However, most existing results related to synchronization of delayed neural networkshave been considered with single time delay [17, 18]. In fact, the differential modelswith multiple time delays unavoidable exist in neural networks. It is worthy to pointout that study about the stability of nonlinear fractional order with multiple time delaysseems quite difficulty. There are only few results on the stability and synchronization offractional-order neural networks with multiple time delays in existing literatures [12,22].Motivated by the above discussions, the aim here is to study the synchronization offractional-order neural networks with multiple time delays. By constructing a Lyapunovfunction, applying an inequality of fractional-order and comparison principles of linearfractional equation with multiple time delays, we obtain some sufficient conditions, whichcan achieve synchronization. The obtained results are novel.The remainder of this article is organized as follows. In Section 2, some definitionsand lemmas are introduced. The model description is also given. In Section 3, the suffi-cient conditions for synchronization are obtained. Numerical simulations are presented inSection 4. Some conclusions are drawn in Section 5.2 Preliminaries and model descriptionThere are some definitions of the fractional-order integrals and derivatives, such asRiemann–Liouville definition and the Caputo definition. From the Laplace transformof fractional derivative, the advantage of the Caputo fractional derivative is that it onlyNonlinear Anal. Model. Control, 22(5):636–645638 W. Zhang et al.requires initial conditions given in terms of integer-order derivatives. Here, the definitionof Caputo derivative is adopted.Definition 1. (See [24].) The fractional integral with noninteger order α > 0 for a func-tion x(t) is defined asIαx(t) =1Γ(α)t∫t0(t− τ)α−1x(τ) dτ,where t > t0, t0 is the initial time, Γ(·) is the gamma function, given by Γ(s) =∫∞0ts−1e−t dt.Definition 2. (See [24].) The Caputo fractional derivative of order α for a function x(t)is defined asDαx(t) =1Γ(n− α)t∫t0(t− τ)n−α−1x(n)(τ) dτ,in which t > t0, t0 is the initial time, n− 1 < α < n ∈ Z+.We now consider the following fractional-order neural networks with time delays asmaster system:Dαxi(t) = −cixi(t) +n∑j=1aijfj(xj(t))+n∑j=1bijgj(xj(t− τj))+ Ii, (1)i, j = 1, 2, . . . , n, where 0 < α < 1, n is the number of units in a neural network,xi(t) denotes the pseudostate variable of the ith unit of master system, ci > 0 is the self-regulating parameters of the ith unit. Ii represents the external input of the ith unit, aijand bij denote the strength of the jth unit on the ith unit at time t and t− τj , respectively,τj > 0 is the transmission delay, fj(xj(t)) and gj(xj(t)) denote the output of the jth unitat time t and t− τj , respectively.The slave system is given byDαyi(t) = −ciyi(t) +n∑j=1aijfj(yj(t))+n∑j=1bijgj(yj(t− τj))− ui(t) + Ii, (2)i, j = 1, 2, . . . , n. Note that yi(t) is the pseudostate vector of slave system and ui(t) isa suitable controller.To ensure the main results, we present the following assumption and lemmas.Assumption 1. The neuron activation functions fj(x), gj(x) satisfy the following Lips-chitz conditions with Lipschitz constants lj > 0, hj > 0:∣∣fj(u)− fj(v)∣∣ 6 lj |u− v|, ∣∣gj(u)− gj(v)∣∣ 6 hj |u− v|for all u, v ∈ R.https://www.mii.vu.lt/NASynchronization of a class of fractional-order neural networks 639Lemma 1. (See [1, 11].) Suppose that x(t) ∈ Rn is a continuous and differentiablevector-value function. Then for any time instant t > t0, we have12Dαx2(t) 6 x(t)Dαx(t) (3)when 0 < α < 1.Lemma 2. (See [22].) Suppose that V (t) ∈ R is a continuous, differentiable, and non-negative function satisfyingDαV (t) 6 −aV (t) +n∑j=1bjV (t− τj), 0 < α < 1, 1 6 j 6 n,V (t) = ϕ(t) > 0, t ∈ [−τ, 0].(4)If a >√2∑nj=1 bj and bj > 0, j = 1, 2, . . . , n, then for all ϕ(t) > 0, τj > 0,limt→+∞ V (t) = 0.3 Synchronization of fractional-order neural networks with multipletime delaysWe will discuss the master-slave synchronization of fractional-order neural networks. Thepurpose is to choose an effective controller to achieve the synchronization of the master-slave systems.Let ei(t) = yi(t)− xi(t), i = 1, 2, . . . , n, be the synchronization errors.Select the control input function as follows:ui(t) = ki(yi(t)− xi(t)), i = 1, 2, . . . , n, (5)where ki denotes the controller feedback gain.Then the error systems are obtained in the formDαei(t) = −ciei(t) +n∑j=1aij[fj(yj(t))− fj(xj(t))]+n∑j=1bij[gj(yj(t− τj))− gj(xj(t− τj))]− kiei(t). (6)Theorem 1. Under Assumption 1 and since the control function satisfies (5), if the con-troller feedback gain ki satisfies the inequalityλ >√22n∑j=1bj , (7)then the fractional-order delayed neural networks system (1) synchronizes system (2),where λ = min16i6n{2(ci + ki)−∑nj=1 lj |aij | −∑nj=1 li|aji| −∑nj=1 hj |bij |}, bj =∑ni=1 hj |bij |.Nonlinear Anal. Model. Control, 22(5):636–645640 W. Zhang et al.Proof. We define the following Lyapunov function:V (t) =12eT(t)e(t), e(t) = (e1(t), . . . , en(t))T. (8)From Lemma 1,DαV (t) = Dα[12eT(t)e(t)]= Dα[12n∑i=1e2i (t)]6n∑i=1ei(t)Dαei(t)=n∑i=1{−cie2i (t) + ei(t)n∑j=1aij[fj(yj(t))− fj(xj(t))]+ ei(t)n∑j=1bij[gj(yj(t− τj))− gj(xj(t− τj))]− kie2i (t)}6n∑i=1{−cie2i (t) +n∑j=1∣∣ei(t)∣∣|aij |∣∣fj(yj(t))− fj(xj(t))∣∣+n∑j=1∣∣ei(t)∣∣|bij |∣∣gj(yj(t− τj))− gj(xj(t− τj))∣∣− kie2i (t)}6n∑i=1{−(ci + ki)e2i (t) +n∑j=1∣∣ei(t)∣∣|aij |lj∣∣ej(t)∣∣+n∑j=1∣∣ei(t)∣∣|bij |hj∣∣ej(t− τj)∣∣}. (9)Note that ∣∣ei(t)∣∣|aij |lj∣∣ej(t)∣∣ 6 12lj |aij |(e2i (t) + e2j (t)),∣∣ei(t)∣∣|bij |hj∣∣ej(t− τj)∣∣ 6 12hj |bij |(e2i (t) + e2j (t− τj)),e2j (t− τj) 6 eTj (t− τj)e(t− τj) = 2V (t− τj).Substitute these into (9), one hasDαV (t) 6n∑i=1{−(ci + ki)e2i (t) +12n∑j=1lj |aij |(e2i (t) + e2j (t))+12n∑j=1hj |bij |(e2i (t) + e2j (t− τj))}= −n∑i=1{(ci + ki)− 12n∑j=1lj |aij | − 12n∑j=1li|aji| − 12n∑j=1hj |bij |}e2i (t)https://www.mii.vu.lt/NASynchronization of a class of fractional-order neural networks 641+12n∑i=1n∑j=1hj |bij |e2j (t− τj)6 − min16i6n{2(ci + ki)−n∑j=1lj |aij | −n∑j=1li|aji| −n∑j=1hj |bij |}V (t)+n∑j=1n∑i=1hj |bij |V (t− τj). (10)Letλ = min16i6n{2(ci + ki)−n∑j=1lj |aij | −n∑j=1li|aji| −n∑j=1hj |bij |},bj =n∑i=1hj |bij |,According to Lemma 2, when λ > (√2/2)∑nj=1 bj , system (1) synchronizessystem (2).For convenience, we rewrite (10) asDαV (t) 6 λV (t) +n∑j=1LV (t− τj), (11)where L = max{bj}.According to Theorem 1, the following corollary holds.Corollary 1. Under Assumption 1 and since the control function satisfies (5), if thecontroller feedback gain ki satisfies the inequalityλ >√22n∑j=1L, (12)then the fractional-order delayed neural networks system (1) synchronizes system (2).Remark 1. If τ1 = τ2 = · · · = τj = τ , the model will reduce to system with single timedelay.Remark 2. Here, we have used comparison principles of fractional-order couple sys-tem with multiple time delays. Some synchronization results of fractional-order neuralnetworks with multiple time delays are derived. The method is novel.Remark 3. Generally, the differential models with multiple time delays unavoidable existin neural networks. However, the research of synchronization of fractional-order neuralnetworks with multiple time delays has seldom seen. A comparative study reveals that ourresults are with multiple time delays. Hence, the results obtained in this paper are moregeneral than the existing results dealing with the single time delay [3–5, 10, 12].Nonlinear Anal. Model. Control, 22(5):636–645642 W. Zhang et al.4 Numerical simulationWe now consider the following two-dimensional fractional-order neural networks withtwo time delays as drive system:Dαx1(t) = −c1x1 + a11f1(x1(t))+ a12f2(x2(t))+ b11g1(x1(t− τ1))+ b12g2(x1(t− τ2))+ I1,Dαx2(t) = −c2x2 + a21f1(x1(t))+ a22f2(x2(t))+ b21g1(x1(t− τ1))+ b22g2(x1(t− τ2))+ I2(13)with α = 0.92, c1 = c2 = 1, a11 = 2.0, a12 = −0.1, a21 = −5.0, a22 = 2.0,b11 = −1.5, b12 = −0.1, b21 = −0.2, b11 = −1.5, I1 = I2 = 0, τ1 = 1, τ2 = 2,fi(s) = gi(s) = tanh(s) for s ∈ R. Obviously, fi(xi) and gi(xi) satisfy Assumption 1with li = hi = 1.The response system is given as follows:Dαy1(t) = −c1y1 + a11f1(y1(t))+ a12f2(y2(t))+ b11g1(y1(t− τ1))+ b12g2(y1(t− τ2))+ I1 − k1(y1 − x1),Dαy2(t) = −c2y2 + a21f1(y1(t))+ a22f2(y2(t))+ b21g1(y1(t− τ1))+ b22g2(y1(t− τ2))+ I2 − k2(y2 − x2).(14)If we select the control gain k1 = k2 = 11, by simple computing, the condition ofTheorem 1 is satisfied. We denote the initial values of state vector x1(t), x2(t), y1(t),y2(t) of the master-slave systems as follows: x˜1(t), x˜2(t), y˜1(t), y˜2(t), select the initialvalues in such a way that when t ∈ [−2, 0], x˜1(t) = 10 tanh(pi(t + 2)/2), x˜2(t) =10 cos(pi(t + 2)/2), y˜1(t) = 10(pi(t + 2)/2), y˜2(t) = 10(pi(t + 2)/2). Under theseparameters, the state trajectories of system without controller and with the control gainare shown in Fig. 1. The synchronization errors are shown in Fig. 2. The state synchro-nization trajectories of master-slave systems are shown in Fig. 3.Figure 1. Trajectories of the system without controller (13) and with the control gain (14).https://www.mii.vu.lt/NASynchronization of a class of fractional-order neural networks 643Figure 2. Synchronization error trajectories of ei, i = 1, 2.Figure 3. State synchronization trajectories of xi, yi, i = 1, 2.5 ConclusionsHere, we investigated the synchronization of fractional-order neural networks with mul-tiple time delays. By using Lyapunov function and the comparison principles of linearfractional equation with multiple time delays, some sufficient conditions are derived toensure the synchronization of the master-slave systems. A numerical example is presentedto verify the effectiveness of the theoretical results.Acknowledgment. We would like to thank the referees and the editor for their valuablecomments.References1. N. Aguila-Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractionalorder systems, Commun. Nonlinear Sci. Numer. Simul., 19:2951–2957, 2014.2. E. Ahmed, A. Elgazzar, On fractional order differential equations model for nonlocalepidemics, Physica A, 379:607–614, 2007.Nonlinear Anal. Model. Control, 22(5):636–645644 W. Zhang et al.3. H.B. Bao, J.D. Cao, Projective synchronization of fractional-order memristor-based neuralnetworks, Neural Netw., 63:1–9, 2015.4. H.B. Bao, J. H. Park, J.D. Cao, Synchronization of fractional-order complex-valued neuralnetworks with time delay, Neural Netw., 81:16–28, 2016.5. H.B. Bao, J.H. Park, J.D. Cao, Adaptive synchronization of fractional-order memristor-basedneural networks with time delay, Nonlinear Dyn., 158:1343–1354, 2015.6. P. Butzer, U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore,2000.7. J.D. Cao, R.X. Li, Fixed-time synchronization of delayed memristor-based recurrent neuralnetworks, Sci. China, Inf. Sci., 60:032201, 2017.8. J.D. Cao, R. Rakkiyappan, K. Maheswari, A. Chandrasekar, ExponentialH∞ filtering analysisfor discrete-time switched neural networks with random delays using sojourn probabilities, Sci.China, Technol. Sci., 59:387–402, 2016.9. J. Chen, G. Zeng, P. Jiang, Global Mittag–Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural Netw., 51:1–8, 2014.10. L.P. Chen, R.C. Wu, J.D. Cao, J.B. Liu, Stability and synchronization of memristor-basedfractional-order delayed neural networks, Neural Netw., 71:37–44, 2015.11. M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, Using generalquadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems,Commun. Nonlinear Sci. Numer. Simul., 22:650–659, 2015.12. Y.J. Gu, Y.G. Yu, H. Wang, Synchronization for fractional-order time-delayed memristor-basedneural networks with parameter uncertainty, J. Franklin Inst., 353:3657–3684, 2016.13. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.14. C. Huang, J. Cao, M. Xiao, A. Alsaedi, T. Hayat, Bifurcations in a delayed fractional complex-valued neural network, Appl. Math. Comput., 292:210–227, 2017.15. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional DifferentialEquations, Elsevier, Amsterdam, 2006.16. N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268:298–305,2000.17. X. Li, X. Fu, Lag synchronization of chaotic delayed neural networks via impulsive control,IMA J. Math. Control Inf., 29:133–145, 2012.18. X. Li, R. Rakkiyappan, N. Sakthivel, Non-fragile synchronization control for Markovianjumping complex dynamical networks with probabilistic time-varying coupling delay, Asian J.Control, 17:1678–1695, 2015.19. X. Li, S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEETrans. Autom. Control, 62:406–411, 2017.20. X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses,Automatica, 64:63–69, 2016.21. X. Li, X. Zhang, S. Song, Effect of delayed impulses on input-to-state stability of nonlinearsystems, Automatica, 76:378–382, 2017.22. S. Liang, R.C. Wu, L.P. Chen, Comparison principles and stability of nonlinear fractional-ordercellular neural networks with multiple time delays, Neurocomputing, 168:618–625, 2015.https://www.mii.vu.lt/NASynchronization of a class of fractional-order neural networks 64523. N. Özalp, E. Demirci, A fractional order SEIR model with vertical transmission, Math.Comput. Modelling, 54:1–6, 2011.24. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.25. I. Stamova, Global Mittag–Leffler stability and synchronization of impulsive fractional-orderneural networks with time-varying delays, Nonlinear Dyn., 77:1251–1260, 2014.26. G. Velmurugan, R. Rakkiyappan, J.D. Cao, Finite-time synchronization of fractional-ordermemristor-based neural-networks with time delays, Neural Netw., 73:36–46, 2016.27. F. Wang, Y. Yang, M. Hu, X. Xu, Projective cluster synchronization of fractional-ordercoupled-delay complex network via adaptive pinning control, Physica A, 434:134–143, 2015.28. Y.L. Wang, J.D. Cao, Cluster synchronization in nonlinearly coupled delayed networks ofnon-identical dynamic systems, Nonlinear Anal., Real World Appl., 14:842–851, 2013.29. Y.Q. Yang, J.D. Cao, Exponential synchronization of the complex dynamical networks witha coupling delay and impulsive effects, Nonlinear Anal., Real World Appl., 11:1650–1659,2010.Nonlinear Anal. Model. Control, 22(5):636–645

Synchronization of a class of fractional-order neural networks with multiple time delays by comparison principles

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Synchronization of a class of fractional-order neural networks with multiple time delays by comparison principles

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Nonlinear Analysis: Modelling and Control