Variable-order fractional discrete-time recurrent neural networks

Abstract Discrete fractional calculus is suggested to describe neural networks with memory effects. Fractional discrete-time recurrent neural network is proposed on an isolated time scale. Stability results are investigated via Banach fixed point technique. The attractive solution space is constructed and stability conditions are provided. Furthermore, short memory and variable-order fractional neural networks are given according to the stability conditions. Two and three dimensional numerical examples are used to demonstrate the theoretical results.

[1]  Yongguang Yu,et al.  Mittag-Leffler stability of fractional-order Hopfield neural networks , 2015 .

[2]  S. Sengul.  Discrete Fractional Calculus and Its Applications to Tumor Growth , 2010 .

[3]  Raghib Abu-Saris,et al.  On the asymptotic stability of linear system of fractional-order difference equations , 2013 .

[4]  Tiedong Ma,et al.  Dynamic analysis of a class of fractional-order neural networks with delay , 2013, Neurocomputing.

[5]  Fulai Chen,et al.  Fixed points and asymptotic stability of nonlinear fractional difference equations , 2011 .

[6]  Michael T. Holm,et al.  The Laplace transform in discrete fractional calculus , 2011, Comput. Math. Appl..

[7]  Guo-Cheng Wu,et al.  New variable-order fractional chaotic systems for fast image encryption. , 2019, Chaos.

[8]  Dumitru Baleanu,et al.  Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique , 2019, Nonlinear Analysis: Modelling and Control.

[9]  Thabet Abdeljawad,et al.  On Riemann and Caputo fractional differences , 2011, Comput. Math. Appl..

[10]  Delfim F. M. Torres,et al.  Discrete-time fractional variational problems , 2010, Signal Process..

[11]  D. Baleanu,et al.  Discrete fractional logistic map and its chaos , 2014 .

[12]  Zhigang Zeng,et al.  Global Mittag–Leffler Stabilization of Fractional-Order Memristive Neural Networks , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[13]  Zhigang Zeng,et al.  Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks , 2014, Neural Networks.

[14]  Jinde Cao,et al.  State estimation of fractional-order delayed memristive neural networks , 2018, Nonlinear Dynamics.

[15]  F. Atici,et al.  Modeling with fractional difference equations , 2010 .

[16]  Zhen Wang,et al.  Chaos and hyperchaos in fractional-order cellular neural networks , 2012, Neurocomputing.

[17]  Ahmed S Elwakil,et al.  Fractional-order circuits and systems: An emerging interdisciplinary research area , 2010, IEEE Circuits and Systems Magazine.

[18]  Jiang Wei,et al.  Solving Fractional Difference Equations Using the Laplace Transform Method , 2014 .

[19]  Eva Kaslik,et al.  Nonlinear dynamics and chaos in fractional-order neural networks , 2012, Neural Networks.

[20]  T. Furumochi,et al.  Krasnoselskii's fixed point theorem and stability , 2002 .

[21]  I. Podlubny.  Fractional differential equations , 1998 .

[22]  I. Stamova.  Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays , 2014, Nonlinear Dynamics.

[23]  P. Eloe,et al.  Initial value problems in discrete fractional calculus , 2008 .

[25]  A. Peterson,et al.  Discrete Fractional Calculus , 2016 .

[26]  George A. Anastassiou.  About Discrete Fractional Calculus with Inequalities , 2011 .

[27]  J. Cermák,et al.  On explicit stability conditions for a linear fractional difference system , 2015 .

[28]  S. Westerlund,et al.  Capacitor theory , 1994 .

[29]  Jinde Cao,et al.  Non-fragile state estimation for delayed fractional-order memristive neural networks , 2019, Appl. Math. Comput..