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

Volume 18, Issue 3, pp 328--345 Publication Date: June 27, 2018       Article History
• 490 Views

### 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.

### Keywords

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

•  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.

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

• [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.

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

• [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.

• [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.

• [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.

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

• [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.

• [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.

• [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.

• [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.

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

• [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.

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

• [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.

• [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.

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

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

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

• [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.

• [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.

• [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.

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

• [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.

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

• [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.

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

• [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.

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

• [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.

• [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.

• [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.