%0 Journal Article %T Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays %A Agarwal, R. P. %A Hristova, S. %A O'Regan, Donal %J Journal of Mathematics and Computer Science %D 2018 %V 18 %N 3 %@ ISSN 2008-949X %F Agarwal2018 %X 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. %9 journal article %R 10.22436/jmcs.018.03.08 %U http://dx.doi.org/10.22436/jmcs.018.03.08 %P 328--345 %0 Journal Article %T Lyapunov functions and strict stability of Caputo fractional differential equations %A R. Agarwal %A S. Hristova %A D. O’Regan %J Adv. Difference Equ. %D 2015 %V 2015 %F Agarwal2015 %0 Journal Article %T Lyapunov functions and stability of Caputo fractional differential equations with delays %A R. Agarwal %A S. Hristova %A D. O’Regan %J %D to be published %V %F Agarwalto be published %0 Journal Article %T A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations %A R. Agarwal %A S. Hristova %A D. O’Regan %J Fract. Calc. Appl. Anal. %D 2016 %V 19 %F Agarwal2016 %0 Journal Article %T Stability of Caputo fractional differential equations by Lyapunov functions %A R. Agarwal %A D. O’Regan %A S. Hristova %J Appl. Math. %D 2015 %V 60 %F Agarwal2015 %0 Journal Article %T Practical stability with respect to initial time difference for Caputo fractional differential equations %A R. Agarwal %A D. O’Regan %A S. Hristova %A M. Cicek %J Commun. Nonlinear Sci. Numer. Simul. %D 2017 %V 42 %F Agarwal2017 %0 Journal Article %T Synchronization of fractional-order complex-valued neural networks with time delay %A H. Bao %A J. H. Park %A J. Cao %J Neural Netw. %D 2016 %V 81 %F Bao2016 %0 Journal Article %T Dynamic analysis of a class of fractional-order neural networks with delay %A L. Chen %A Y. Chai %A R. Wu %A T. Ma %A H. Zhai %J Neurocomputing %D 2013 %V 111 %F Chen2013 %0 Journal Article %T Razumikhin-type stability theorems for functional fractional-order differential systems and applications %A B. Chen %A J. Chen %J Appl. Math. Comput. %D 2015 %V 254 %F Chen2015 %0 Journal Article %T Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks %A D. Chen %A R. Zhang %A X. Liu %A X. Ma %J Commun. Nonlinear Sci. Numer. Simul. %D 2014 %V 19 %F Chen2014 %0 Journal Article %T Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems %A M. A. Duarte-Mermoud %A N. Aguila-Camacho %A J. A. Gallegos %A R. Castro-Linares %J Commun. Nonlinear Sci. Numer. Simul. %D 2015 %V 22 %F Duarte-Mermoud2015 %0 Journal Article %T Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions %A Y. Huang %A H. Zhang %A Z. Wang %J Neurocomputing %D 2012 %V 91 %F Huang2012 %0 Journal Article %T Stability analysis of fractional-order delayed neural networks %A R. Li %A J. Cao %A A. Alsaedi %A F. Alsaadi %J Nonlinear Anal. Model. Control %D 2017 %V 22 %F Li2017 %0 Journal Article %T Mittag-Leffler stability of fractional order nonlinear dynamic systems %A Y. Li %A Y. Chen %A I. Podlubny %J Automatica J. IFAC %D 2009 %V 45 %F Li2009 %0 Journal Article %T Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability %A Y. Li %A Y. Chen %A I. Podlubny %J Comput. Math. Appl. %D 2010 %V 59 %F Li2010 %0 Journal Article %T Delay-interval-dependent stability of recurrent neural networks with time-varying delay %A C. Li %A G. Feng %J Neurocomputing %D 2009 %V 72 %F Li2009 %0 Journal Article %T Global Exponential Stability of Reaction-Diffusion Recurrent Neural Networks with Time-Varying Delays %A J. Liang %A J. Cao %J Phys. Lett. A %D 2003 %V 314 %F Liang2003 %0 Journal Article %T Boundedness and Exponential Stability for Nonautonomous Cellular Neural Networks with Reaction- Diffusion Terms %A X. Lou %A B. Cui %J Chaos Solitons Fractals %D 2007 %V 33 %F Lou2007 %0 Journal Article %T Stability of analog neural networks with delay %A C. M. Marcus %A R. M. Westervelt %J Phys. Rev. A %D 1989 %V 39 %F Marcus1989 %0 Journal Article %T Communities, modules and large-scale structure in networks %A M. E. J. Newman %J Nature Phys. %D 2012 %V 8 %F Newman2012 %0 Book %T Fractional Differential Equations %A I. Podlubny %D 1999 %I Academic Press %C San Diego %F Podlubny1999 %0 Journal Article %T Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning %A A. Rahimi %A B. Recht %J Adv. Neural Information Processing Syst. %D 2008 %V 2008 %F Rahimi2008 %0 Journal Article %T Some fractional comparison results and stability theorem for fractional time delay systems %A S. J. Sadati %A R. Ghaderi %A A. Ranjbar %J Rom. Reports Phy. %D 2013 %V 65 %F Sadati2013 %0 Journal Article %T Space-time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann-Liouville Fractional Derivative %A R. K. Saxena %A A. M. Mathai %A H. J. Haubold %J Axioms %D 2014 %V 3 %F Saxena2014 %0 Journal Article %T On the Lyapunov theory for functional differential equations of fractional order %A I. M. Stamova %J Proc. Amer. Math. Soc. %D 2016 %V 144 %F Stamova2016 %0 Journal Article %T Delayed ReactionDiffusion Cellular Neural Networks of Fractional Order: MittagLeffler Stability and Synchronization %A I. M. Stamova %A S. Simeonov %J J. Comput. Nonlinear Dynam. %D 2017 %V 13 %F Stamova2017 %0 Book %T Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications %A I. Stamova %A G. Stamov %D 2016 %I CRC Press %C New York %F Stamova2016 %0 Journal Article %T The complex folding network of single calmodulin molecules %A J. Stigler %A F. Ziegler %A A. Gieseke %A J. C. M. Gebhardt %A M. Rief %J Science %D 2011 %V 334 %F Stigler2011 %0 Journal Article %T Variational Lyapunov method for fractional differential equations %A J. Vasundhara Devi %A F. A. Mc Rae %A Z. Drici %J Comput. Math. Appl. %D 2012 %V 64 %F Devi2012 %0 Journal Article %T Global properties of a two-scale network stochastic delayed human epidemic dynamic model %A D. Wanduku %A G. S. Ladde %J Nonlinear Anal. Real World Appl. %D 2012 %V 13 %F Wanduku2012 %0 Journal Article %T Stability analysis of fractional-order Hopfield neural networks with time delays %A H. Wang %A Y. Yu %A G. Wen %J Neural Netw. %D 2014 %V 55 %F Wang2014 %0 Journal Article %T Exponential stability of impulsive discrete systems with time delay and applications in stochastic neural networks: a Razumikhin approach %A S. Wu %A C. Li %A X. Liao %A S. Duan %J Neurocomputing %D 2012 %V 82 %F Wu2012 %0 Journal Article %T Finite-time stability analysis of fractional-order neural networks with delay %A X. Yang %A Q. Song %A Y. Liu %A Z. Zhao %J Neurocomputing %D 2015 %V 152 %F Yang2015 %0 Journal Article %T Synchronization of a class of fractional-order neural networks with multiple time delays by comparison principles %A W. Zhang %A R. Wu %A J. Cao %A A. Alsaedi %A T. Hayat %J Nonlinear Anal. Model. Control %D 2017 %V 22 %F Zhang2017