This research introduces a model of a delayed reaction–diffusion fractional neural network with time-varying delays. The Mittag–Leffler-type stability of the solutions is investigated, and new sufficient conditions are established by the use of the fractional Lyapunov method. Mittag–Leffler-type synchronization criteria are also derived. Three illustrative examples are established to exhibit the proposed sufficient conditions.
Issue Section:
Research Papers
References
1.
Chua
, L. O.
, and Yang
, L.
, 1988
, “Cellular Neural Networks: Theory
,” IEEE Trans. Circuits Syst.
, 35
(10
), pp. 1257
–1272
.2.
Chua
, L. O.
, and Yang
, L.
, 1988
, “Cellular Neural Networks: Applications
,” IEEE Trans. Circuits Syst.
, 35
(10
), pp. 1273
–1290
.3.
Chua
, L. O.
, 1999
, “Passivity and Complexity
,” IEEE Trans. Circuits Syst. I
, 46
(1
), pp. 71
–82
.4.
Itoh
, M.
, and Chua
, L. O.
, 2006
, “Complexity of Reaction-Diffusion CNN
,” Internat. J. Bifur. Chaos
, 16
(9
), pp. 2499
–2527
.5.
Liang
, J. L.
, and Cao
, J.
, 2003
, “Global Exponential Stability of Reaction-Diffusion Recurrent Neural Networks With Time-Varying Delays
,” Phys. Lett. A
, 314
(5–6
), pp. 434
–442
.6.
Lou
, X.
, and Cui
, B.
, 2007
, “Boundedness and Exponential Stability for Nonautonomous Cellular Neural Networks With Reaction-Diffusion Terms
,” Chaos Solitons Fractals
, 33
(2
), pp. 653
–662
.7.
Lu
, J. G.
, 2008
, “Global Exponential Stability and Periodicity of Reaction-Diffusion Delayed Recurrent Neural Networks With Dirichlet Boundary Conditions
,” Chaos Solitons Fractals
, 35
(1
), pp. 116
–125
.8.
Wang
, Y.
, and Cao
, J.
, 2007
, “Synchronization of a Class of Delayed Neural Networks With Reaction-Diffusion Terms
,” Phys. Lett. A
, 369
(3
), pp. 201
–211
.9.
Chen
, W. H.
, Liu
, L.
, and Lu
, X.
, 2017
, “Intermittent Synchronization of Reaction-Diffusion Neural Networks With Mixed Delays Via Razumikhin Technique
,” Nonlinear Dyn.
, 87
(1
), pp. 535
–551
.10.
Gan
, Q.
, 2017
, “Exponential Synchronization of Generalized Neural Networks With Mixed Time-Varying Delays and Reaction-Diffusion Terms Via Aperiodically Intermittent Control
,” Chaos
, 27
(1
), p. 013113
.11.
Rakkiyappan
, R.
, Dharani
, S.
, and Zhu
, Q.
, 2015
, “Synchronization of Reaction-Diffusion Neural Networks With Time-Varying Delays Via Stochastic Sampled-Data Controller
,” Nonlinear Dyn.
, 79
(1
), pp. 485
–500
.12.
Diethelm
, K.
, 2010
, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type
, Springer
, Berlin
.13.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, Theory and Applications of Fractional Differential Equations
, Elsevier
, Amsterdam, The Netherlands
.14.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, San Diego, CA
. 15.
Trigeassou
, J. C.
, Maamri
, N.
, and Oustaloup
, A.
, 2016
, “Lyapunov Stability of Commensurate Fractional Order Systems: A Physical Interpretation
,” ASME J. Comput. Nonlinear Dyn.
, 11
(5
), p. 051007
.16.
Baleanu
, D.
, Diethelm
, K.
, Scalas
, E.
, and Trujillo
, J.
, 2017
, Fractional Calculus: Models and Numerical Methods
, World Scientific
, Hackensack, NJ
.17.
Baleanu
, D.
, and Mustafa
, O. G.
, 2015
, Asymptotic Integration and Stability for Ordinary, Functional and Discrete Differential Equations of Fractional Order
, World Scientific
, Hackensack, NJ
.18.
Stamova
, I.
, and Stamov
, G.
, 2013
, “Lipschitz Stability Criteria for Functional Differential Systems of Fractional Order
,” J. Math. Phys.
, 54
(4), p. 043502
.19.
Stamova
, I.
, and Stamov
, G.
, 2017
, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications
, CRC Press
, Boca Raton, FL
.20.
Gafiychuk
, V.
, and Datsko
, B.
, 2010
, “Mathematical Modeling of Different Types of Instabilities in Time Fractional Reaction-Diffusion Systems
,” Comput. Math. Appl.
, 59
(3
), pp. 1101
–1107
.21.
Lenzi
, E. K.
, dos Santos
, M. A. F.
, Lenzi
, M. K.
, and Menechini Neto
, R.
, 2017
, “Solutions for a Mass Transfer Process Governed by Fractional Diffusion Equations With Reaction Terms
,” Commun. Nonlinear Sci. Numer. Simul.
, 48
, pp. 307
–317
.22.
Ouyang
, Z.
, 2011
, “Existence and Uniqueness of the Solutions for a Class of Nonlinear Fractional Order Partial Differential Equations With Delay
,” Comput. Math. Appl.
, 61
(4
), pp. 860
–870
.23.
Zhu
, B.
, Liu
, L.
, and Wu
, Y.
, 2016
, “Local and Global Existence of Mild Solutions for a Class of Nonlinear Fractional Reaction-Diffusion Equations With Delay
,” Appl. Math. Lett.
, 61
, pp. 73
–79
.24.
Chen
, B. S.
, and Chen
, J. J.
, 2015
, “Global Asymptotical ω–Periodicity of a Fractional-Order Non-Autonomous
,” Neural Networks
, 68
, pp. 78
–88
.25.
Kaslik
, E.
, and Sivasundaram
, S.
, 2012
, “Nonlinear Dynamics and Chaos in Fractional Order Neural Networks
,” Neural Networks
, 32
, pp. 245
–256
.26.
Song
, C.
, and Cao
, J.
, 2014
, “Dynamics in Fractional-Order Neural Networks
,” Neurocomputing
, 142
, pp. 494
–498
.27.
Rakkiyappan
, R.
, Velmurugan
, G.
, and Cao
, J.
, 2015
, “Stability Analysis of Memristor-Based Fractional-Order Neural Networks With Different Memductance Functions
,” Cognit. Neurodynamics
, 9
(2
), pp. 145
–177
.28.
Lundstrom
, B.
, Higgs
, M.
, Spain
, W.
, and Fairhall
, A.
, 2008
, “Fractional Differentiation by Neocortical Pyramidal Neurons
,” Nat. Neurosci.
, 11
, pp. 1335
–1342
.29.
Stamov
, G.
, and Stamova
, I.
, 2017
, “Impulsive Fractional-Order Neural Networks With Time-Varying Delays: Almost Periodic Solutions
,” Neural Comput. Appl.
, 28
(11
), pp. 3307
–3316
.30.
Wang
, H.
, Yu
, Y.
, Wen
, G.
, and Zhang
, S.
, 2015
, “Stability Analysis of Fractional-Order Neural Networks With Time Delay
,” Neural Process. Lett.
, 42
(2
), pp. 479
–500
.31.
Wu
, R.
, Hei
, X.
, and Chen
, L.
, 2013
, “Finite-Time Stability of Fractional-Order Neural Networks With Delay
,” Commun. Theor. Phys.
, 60
(2
), pp. 189
–193
.32.
Ke
, Y.
, and Miao
, C.
, 2015
, “Stability Analysis of Fractional-Order Cohen-Grossberg Neural Networks With Time Delay
,” Int. J. Comput. Math.
, 92
(6
), pp. 1102
–1113
.33.
Golmankhaneh
, A. K.
, Arefi
, R.
, and Baleanu
, D.
, 2015
, “Synchronization in a Non-Identical Fractional Order of a Proposed Modified System
,” J. Vib. Control
, 21
(6
), pp. 1154
–1161
.34.
Singh
, A. K.
, Yadav
, V. K.
, and Das
, S.
, 2016
, “Dual Combination Synchronization of the Fractional Order Complex Chaotic Systems
,” ASME J. Comput. Nonlinear Dyn.
, 12
(1
), p. 011017
.35.
Velmurugan
, G.
, and Rakkiyapan
, R.
, 2015
, “Hybrid Projective Synchronization of Fractional Order Chaotic Complex Systems With Time Delays
,” ASME J. Comput. Nonlinear Dyn.
, 11
(3
), p. 031016
.36.
Wu
, G. C.
, Baleanu
, D.
, Xie
, H. P.
, and Chen
, F. L.
, 2016
, “Chaos Synchronization of Fractional Chaotic Maps Based on the Stability Condition
,” Phys. A
, 460
, pp. 374
–383
.37.
Zhou
, Q.
, and Wan
, L.
, 2009
, “Impulsive Effects on Stability of Cohen-Grossberg-Type Bidirectional Associative Memory Neural Networks With Delays
,” Nonlinear Anal. Real World Appl.
, 10
(4
), pp. 2531
–2540
.38.
Li
, Y.
, Chen
, Y. Q.
, and Podlubny
, I.
, 2010
, “Stability of Fractional Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag-Leffler Stability
,” Comput. Math. Appl.
, 59
(5
), pp. 1810
–1821
.39.
Stamova
, I. M.
, 2015
, “Mittag-Leffler Stability of Impulsive Differential Equations of Fractional Order
,” Q. Appl. Math.
, 73
, pp. 525
–535
.40.
Chen
, J. J.
, Zeng
, Z. G.
, and Jiang
, P.
, 2014
, ” “Global Mittag-Leffler Stability and Synchronization of Memristor-Based Fractional-Order Neural Networks
,” Neural Networks
, 51
, pp. 1
–8
.41.
Stamova
, I. M.
, 2014
, ” “Global Mittag-Leffler Stability and Synchronization of Impulsive Fractional-Order Neural Networks With Time-Varying Delays
,” Nonlinear Dyn.
, 77
(4
), pp. 1251
–1260
.42.
Aguila-Camacho
, N.
, Duarte-Mermoud
, M. A.
, and Gallegos
, J. A.
, 2014
, “Lyapunov Functions for Fractional Order Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 19
(9
), pp. 2951
–2957
.Copyright © 2018 by ASME
You do not currently have access to this content.
