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Research Papers

Delayed Reaction–Diffusion Cellular Neural Networks of Fractional Order: Mittag–Leffler Stability and Synchronization

[+] Author and Article Information
Ivanka M. Stamova

Department of Mathematics,
University of Texas at San Antonio,
San Antonio, TX 78249
e-mail: ivanka.stamova@utsa.edu

Stanislav Simeonov

Department of Computer
Systems and Technologies,
“Prof. Dr. Assen Zlatarov” University,
Burgas 8010, Bulgaria
e-mail: stanislav_simeonov@btu.bg

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 22, 2017; final manuscript received October 19, 2017; published online November 17, 2017. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(1), 011015 (Nov 17, 2017) (7 pages) Paper No: CND-17-1274; doi: 10.1115/1.4038290 History: Received June 22, 2017; Revised October 19, 2017

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.

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References

Chua, L. O. , and Yang, L. , 1988, “ Cellular Neural Networks: Theory,” IEEE Trans. Circuits Syst., 35(10), pp. 1257–1272. [CrossRef]
Chua, L. O. , and Yang, L. , 1988, “ Cellular Neural Networks: Applications,” IEEE Trans. Circuits Syst., 35(10), pp. 1273–1290. [CrossRef]
Chua, L. O. , 1999, “ Passivity and Complexity,” IEEE Trans. Circuits Syst. I, 46(1), pp. 71–82. [CrossRef]
Itoh, M. , and Chua, L. O. , 2006, “ Complexity of Reaction-Diffusion CNN,” Internat. J. Bifur. Chaos, 16(9), pp. 2499–2527. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
Diethelm, K. , 2010, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin.
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA. [PubMed] [PubMed]
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. [CrossRef]
Baleanu, D. , Diethelm, K. , Scalas, E. , and Trujillo, J. , 2017, Fractional Calculus: Models and Numerical Methods, World Scientific, Hackensack, NJ.
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. [CrossRef]
Stamova, I. , and Stamov, G. , 2013, “ Lipschitz Stability Criteria for Functional Differential Systems of Fractional Order,” J. Math. Phys., 54(4), p. 043502. [CrossRef]
Stamova, I. , and Stamov, G. , 2017, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, CRC Press, Boca Raton, FL.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Chen, B. S. , and Chen, J. J. , 2015, “ Global Asymptotical ω–Periodicity of a Fractional-Order Non-Autonomous,” Neural Networks, 68, pp. 78–88. [CrossRef] [PubMed]
Kaslik, E. , and Sivasundaram, S. , 2012, “ Nonlinear Dynamics and Chaos in Fractional Order Neural Networks,” Neural Networks, 32, pp. 245–256. [CrossRef] [PubMed]
Song, C. , and Cao, J. , 2014, “ Dynamics in Fractional-Order Neural Networks,” Neurocomputing, 142, pp. 494–498. [CrossRef]
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. [CrossRef]
Lundstrom, B. , Higgs, M. , Spain, W. , and Fairhall, A. , 2008, “ Fractional Differentiation by Neocortical Pyramidal Neurons,” Nat. Neurosci., 11, pp. 1335–1342. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Stamova, I. M. , 2015, “ Mittag-Leffler Stability of Impulsive Differential Equations of Fractional Order,” Q. Appl. Math., 73, pp. 525–535. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The chaotic behavior of the neural network (17) described in Example 3. Evolution of v1(t, x).

Grahic Jump Location
Fig. 2

The chaotic behavior of the neural network (17) described in Example 3. Evolution of v2(t, x).

Grahic Jump Location
Fig. 3

The synchronization error e(t, x) between systems (17) and (18) described in Example 3. Evolution of e1(t, x).

Grahic Jump Location
Fig. 4

The synchronization error e(t, x) between systems (17) and (18) described in Example 3. Evolution of e2(t, x).

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