Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays

Volume 18, Issue 3, pp 328--345 http://dx.doi.org/10.22436/jmcs.018.03.08

Publication Date: June 27, 2018 Submission Date: November 05, 2017 Revision Date: March 12, 2018 Accteptance Date: April 20, 2018

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Authors

R. P. Agarwal - Department of Mathematics, Texas A\(\&\)M University-Kingsville, Kingsville, TX 78363, USA \(\&\) Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA S. Hristova - University of Plovdiv Paisii Hilendarski, Plovdiv, Bulgaria Donal O'Regan - School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Abstract

A reaction diffusion equation with a Caputo fractional derivative in time and with time-varying delays is considered. Stability properties of the solutions are studied via the direct Lyapunov method and arbitrary Lyapunov functions (usually quadratic Lyapunov functions are used). In this paper we give a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of reaction-diffusion fractional neural network with variable coefficients and time-varying delays. We show the quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability are obtained and we illustrate our theory on a particular nonlinear Caputo reaction-diffusion fractional neural network with time dependent delays.

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Keywords

• Reaction-diffusion fractional neural networks
• delays
• Caputo derivatives
• Lyapunov functions
• stability
• fractional derivative of Lyapunov functions

MSC

•  35R11
•  34K37
•  34K20

References

• [1] R. Agarwal, S. Hristova, D. O’Regan, Lyapunov functions and strict stability of Caputo fractional differential equations, Adv. Difference Equ., 2015 (2015), 20 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] R. Agarwal, S. Hristova, D. O’Regan, Lyapunov functions and stability of Caputo fractional differential equations with delays, , (to be published),
  • View Article
  • Google Scholar


• [3] R. Agarwal, S. Hristova, D. O’Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 290–318.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] R. Agarwal, D. O’Regan, S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60 (2015), 653–676.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] R. Agarwal, D. O’Regan, S. Hristova, M. Cicek, Practical stability with respect to initial time difference for Caputo fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 106–120.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] H. Bao, J. H. Park, J. Cao, Synchronization of fractional-order complex-valued neural networks with time delay, Neural Netw., 81 (2016), 16–28.
  • View Article
  • Google Scholar


• [7] L. Chen, Y. Chai, R. Wu, T. Ma, H. Zhai, Dynamic analysis of a class of fractional-order neural networks with delay, Neurocomputing, 111 (2013), 190–194.
  • View Article
  • Google Scholar


• [8] B. Chen, J. Chen, Razumikhin-type stability theorems for functional fractional-order differential systems and applications, Appl. Math. Comput., 254 (2015), 63–69.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] D. Chen, R. Zhang, X. Liu, X. Ma, Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 4105–4121.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 650–659.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] Y. Huang, H. Zhang, Z. Wang, Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions, Neurocomputing, 91 (2012), 21–28.
  • View Article
  • Google Scholar


• [12] R. Li, J. Cao, A. Alsaedi, F. Alsaadi, Stability analysis of fractional-order delayed neural networks, Nonlinear Anal. Model. Control, 22 (2017), 505–520.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] Y. Li, Y. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC, 45 (2009), 1965–1969.
  • View Article
  • Google Scholar


• [14] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] C. Li, G. Feng, Delay-interval-dependent stability of recurrent neural networks with time-varying delay, Neurocomputing, 72 (2009), 1179–1183.
  • View Article
  • Google Scholar


• [16] J. Liang, J. Cao, Global Exponential Stability of Reaction-Diffusion Recurrent Neural Networks with Time-Varying Delays, Phys. Lett. A, 314 (2003), 434–442.
  • View Article
  • Google Scholar
  • MATH


• [17] X. Lou, B. Cui, Boundedness and Exponential Stability for Nonautonomous Cellular Neural Networks with Reaction- Diffusion Terms, Chaos Solitons Fractals, 33 (2007), 653–662.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] C. M. Marcus, R. M. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A, 39 (1989), 347–359.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] M. E. J. Newman, Communities, modules and large-scale structure in networks, Nature Phys., 8 (2012), 25–31.
  • View Article
  • Google Scholar


• [20] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)
  • Google Scholar


• [21] A. Rahimi, B. Recht, Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning, Adv. Neural Information Processing Syst., 2008 (2008), 1313–1320.
  • Google Scholar


• [22] S. J. Sadati, R. Ghaderi, A. Ranjbar, Some fractional comparison results and stability theorem for fractional time delay systems, Rom. Reports Phy., 65 (2013), 94–102.
  • Google Scholar


• [23] R. K. Saxena, A. M. Mathai, H. J. Haubold, Space-time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann-Liouville Fractional Derivative, Axioms, 3 (2014), 320–334.
  • View Article
  • Google Scholar
  • MATH


• [24] I. M. Stamova, On the Lyapunov theory for functional differential equations of fractional order, Proc. Amer. Math. Soc., 144 (2016), 1581–1593.
  • View Article
  • MathSciNet
  • Google Scholar


• [25] I. M. Stamova, S. Simeonov, Delayed ReactionDiffusion Cellular Neural Networks of Fractional Order: MittagLeffler Stability and Synchronization, J. Comput. Nonlinear Dynam., 13 (2017), 7 pages.
  • Google Scholar


• [26] I. Stamova, G. Stamov, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, CRC Press, New York (2016)
  • Google Scholar
  • MATH


• [27] J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt, M. Rief, The complex folding network of single calmodulin molecules, Science, 334 (2011), 512–516.
  • View Article
  • Google Scholar


• [28] J. Vasundhara Devi, F. A. Mc Rae, Z. Drici, Variational Lyapunov method for fractional differential equations, Comput. Math. Appl., 64 (2012), 2982–2989.
  • View Article
  • MathSciNet
  • Google Scholar


• [29] D. Wanduku, G. S. Ladde, Global properties of a two-scale network stochastic delayed human epidemic dynamic model, Nonlinear Anal. Real World Appl., 13 (2012), 794–816.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] H. Wang, Y. Yu, G. Wen, Stability analysis of fractional-order Hopfield neural networks with time delays, Neural Netw., 55 (2014), 98–109.
  • View Article
  • Google Scholar
  • MATH


• [31] S. Wu, C. Li, X. Liao, S. Duan, Exponential stability of impulsive discrete systems with time delay and applications in stochastic neural networks: a Razumikhin approach, Neurocomputing, 82 (2012), 29–36.
  • View Article
  • Google Scholar


• [32] X. Yang, Q. Song, Y. Liu, Z. Zhao, Finite-time stability analysis of fractional-order neural networks with delay, Neurocomputing, 152 (2015), 19–26.
  • View Article
  • Google Scholar


• [33] W. Zhang, R. Wu, J. Cao, A. Alsaedi, T. Hayat, Synchronization of a class of fractional-order neural networks with multiple time delays by comparison principles, Nonlinear Anal. Model. Control, 22 (2017), 636–645.
  • MathSciNet
  • Google Scholar