Delay-Dependent Stability Criterion of Caputo Fractional Neural Networks with Distributed Delay

This paper is concerned with the finite-time stability of Caputo fractional neural networks with distributed delay. The factors of such systems including Caputo’s fractional derivative and distributed delay are taken into account synchronously. For the Caputo fractional neural network model, a finite-time stability criterion is established by using the theory of fractional calculus and generalized Gronwall-Bellman inequality approach. Both the proposed criterion and an illustrative example show that the stability performance of Caputo fractional distributed delay neural networks is dependent on the time delay and the order of Caputo’s fractional derivative over a finite time.

[1]  Jinde Cao,et al.  Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems , 2013 .

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

[3]  Jinde Cao,et al.  Synchronization of Discontinuous Neural Networks with Delays via Adaptive Control , 2013 .

[4]  Jinde Cao,et al.  General Solution of Linear Fractional Neutral Differential Difference Equations , 2013 .

[5]  Shangbo Zhou,et al.  Chaos control and synchronization in a fractional neuron network system , 2008 .

[6]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[7]  P. Arena,et al.  Chaotic behavior in noninteger-order cellular neural networks , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  M. Lazarevic.  Finite time stability analysis of PDα fractional control of robotic time-delay systems , 2006 .

[9]  Jinde Cao,et al.  Bifurcation and control in a neural network with small and large delays , 2013, Neural Networks.

[10]  Mohammad Bagher Menhaj,et al.  Fractional-Order Hopfield Neural Networks , 2008, ICONIP.

[11]  Wuquan Li,et al.  Finite-time generalized synchronization of chaotic systems with different order , 2011 .

[12]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .

[13]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[14]  George A. Anastassiou,et al.  Fractional neural network approximation , 2012, Comput. Math. Appl..

[15]  A. Fairhall,et al.  Fractional differentiation by neocortical pyramidal neurons , 2008, Nature Neuroscience.

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

[17]  Jinde Cao,et al.  Synchronization of Markovian Coupled Neural Networks With Nonidentical Node-Delays and Random Coupling Strengths , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[18]  V. Lakshmikantham,et al.  Theory of fractional functional differential equations , 2008 .

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

[20]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[21]  Jinde Cao,et al.  Synchronization in an array of nonidentical neural networks with leakage delays and impulsive coupling , 2013, Neurocomputing.

[22]  V. Lakshmikantham,et al.  Basic theory of fractional differential equations , 2008 .

[23]  Aleksandar M. Spasic,et al.  Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach , 2009, Math. Comput. Model..

[24]  Jinde Cao,et al.  Finite-time stochastic synchronization of complex networks , 2010 .

[25]  Shangbo Zhou,et al.  Chaotic synchronization of a fractional neuron network system with time-varying delays , 2009, 2009 International Conference on Communications, Circuits and Systems.

[26]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

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