Mittag‐Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field

Fractional order quaternion‐valued neural networks are a type of fractional order neural networks for which neuron state, synaptic connection strengths, and neuron activation functions are quaternion. This paper is dealing with the Mittag‐Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. The fractional order quaternion‐valued neural networks are separated into four real‐valued systems forming an equivalent four real‐valued fractional order neural networks, which decreases the computational complexity by avoiding the noncommutativity of quaternion multiplication. Via some fractional inequality techniques and suitable Lyapunov functional, a brand new criterion is proposed first to ensure the Mittag‐Leffler stability for the addressed neural networks. Besides, the combination of quaternion‐valued adaptive and impulsive control is intended to realize the asymptotically synchronization between two fractional order quaternion‐valued neural networks. Ultimately, two numerical simulations are provided to check the accuracy and validity of our obtained theoretical results.

[1]  Jinde Cao,et al.  Fixed-time synchronization of quaternion-valued memristive neural networks with time delays , 2019, Neural Networks.

[2]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

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

[4]  Yu Wang,et al.  Mittag-Leffler stability of fractional-order neural networks with impulses , 2016 .

[5]  Chuangxia Huang,et al.  Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting , 2018 .

[6]  Qiankun Song,et al.  Global Mittag-Leffler Synchronization of Fractional-Order Neural Networks Via Impulsive Control , 2017, Neural Processing Letters.

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

[8]  R. Rakkiyappan,et al.  Synchronization of memristor-based delayed BAM neural networks with fractional-order derivatives , 2016, Complex..

[9]  Fuad E. Alsaadi,et al.  Global µ-stability of quaternion-valued neural networks with mixed time-varying delays , 2018, Neurocomputing.

[10]  Haijun Jiang,et al.  Projective synchronization for fractional neural networks , 2014, Neural Networks.

[11]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[12]  Ivo Petras,et al.  Fractional-Order Nonlinear Systems , 2011 .

[13]  Xin Long,et al.  New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays , 2020, Appl. Math. Lett..

[14]  Jinde Cao,et al.  Stability and Hopf Bifurcation of a Delayed Prey-Predator Model with Disease in the Predator , 2019, Int. J. Bifurc. Chaos.

[15]  Chuangxia Huang,et al.  Stability Analysis of SIR Model with Distributed Delay on Complex Networks , 2016, PloS one.

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

[17]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[18]  Jinde Cao,et al.  Fixed time synchronization of delayed quaternion-valued memristor-based neural networks , 2020, Advances in Difference Equations.

[19]  Hui Deng,et al.  Fixed-time synchronization of quaternion-valued neural networks , 2019, Physica A: Statistical Mechanics and its Applications.

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

[21]  Qiankun Song,et al.  Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons , 2018 .

[22]  Vasile Mihai Popov,et al.  Hyperstability of Control Systems , 1973 .

[23]  Lihong Huang,et al.  Stability of Almost Periodic Nicholson’s Blowflies Model Involving Patch Structure and Mortality Terms , 2019, Canadian Mathematical Bulletin.

[24]  Peng Wan,et al.  Impulsive Stabilization and Synchronization of Fractional-Order Complex-Valued Neural Networks , 2019, Neural Processing Letters.

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

[26]  Chaofan Qian,et al.  Novel stability criteria on nonlinear density-dependent mortality Nicholson’s blowflies systems in asymptotically almost periodic environments , 2020 .

[27]  Yongqing Yang,et al.  Lag synchronization for fractional-order memristive neural networks via period intermittent control , 2017, Nonlinear Dynamics.

[28]  Zhixia Ding,et al.  Projective synchronization of nonidentical fractional-order neural networks based on sliding mode controller , 2016, Neural Networks.

[29]  Jinde Cao,et al.  Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays , 2019, Mathematics.

[30]  Jinde Cao,et al.  Almost Periodicity in Impulsive Fractional-Order Reaction–Diffusion Neural Networks With Time-Varying Delays , 2020, IEEE Transactions on Cybernetics.

[31]  Chuangxia Huang,et al.  Dynamics analysis on a class of delayed neural networks involving inertial terms , 2020, Advances in Difference Equations.

[32]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[33]  Chuangxia Huang,et al.  On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities , 2014 .

[34]  Jianquan Lu,et al.  Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays , 2016, Nonlinear Dynamics.

[35]  Jinde Cao,et al.  Stability of antiperiodic recurrent neural networks with multiproportional delays , 2020, Mathematical Methods in the Applied Sciences.

[36]  Lihong Huang,et al.  Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term , 2019, Communications on Pure & Applied Analysis.

[37]  Chuangxia Huang,et al.  Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model , 2017 .

[38]  Lihong Huang,et al.  Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle–focus type , 2019, Nonlinear Analysis: Hybrid Systems.

[39]  Jinde Cao,et al.  Projective synchronization of fractional-order memristor-based neural networks , 2015, Neural Networks.

[40]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[41]  N. Laskin.  Fractional market dynamics , 2000 .

[42]  Lihong Huang,et al.  Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks. , 2019, Mathematical biosciences and engineering : MBE.

[43]  Chuangxia Huang,et al.  Dynamics of a class of delayed reaction–diffusion systems with Neumann boundary condition , 2018 .

[44]  Chuangxia Huang,et al.  Global exponential stability of delayed inertial competitive neural networks , 2020, Advances in Difference Equations.

[45]  Ravi P. Agarwal,et al.  Dynamical behaviors of a food-chain model with stage structure and time delays , 2018, Advances in Difference Equations.

[46]  Andrew Chang.  Quaternionic analysis , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[47]  Jinde Cao,et al.  Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control , 2019, Mathematics.

[48]  E. Ahmed,et al.  On fractional order differential equations model for nonlocal epidemics , 2007, Physica A: Statistical Mechanics and its Applications.

[49]  Zhixia Ding,et al.  Global Mittag-Leffler synchronization of fractional-order neural networks with discontinuous activations , 2016, Neural Networks.

[50]  Zhidong Teng,et al.  Pinning adaptive and impulsive synchronization of fractional-order complex dynamical networks , 2016 .