# zbMATH — the first resource for mathematics

Finite-time synchronization of fractional-order memristor-based neural networks with time delays. (English) Zbl 1398.34110
Summary: In this paper, we consider the problem of finite-time synchronization of a class of fractional-order memristor-based neural networks (FMNNs) with time delays and investigated it potentially. By using Laplace transform, the generalized Gronwall’s inequality, Mittag-Leffler functions and linear feedback control technique, some new sufficient conditions are derived to ensure the finite-time synchronization of addressing FMNNs with fractional order $$\alpha:1<\alpha<2$$ and $$0<\alpha<1$$. The results from the theory of fractional-order differential equations with discontinuous right-hand sides are used to investigate the problem under consideration. The derived results are extended to some previous related works on memristor-based neural networks. Finally, three numerical examples are presented to show the effectiveness of our proposed theoretical results.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34D06 Synchronization of solutions to ordinary differential equations 34K37 Functional-differential equations with fractional derivatives 34A36 Discontinuous ordinary differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text:
##### References:
 [1] Abdurahman, A.; Jiang, H.; Teng, Z., Finite-time synchronization for memristor-based neural networks with time-varying delays, Neural Networks, 69, 20-28, (2015) · Zbl 1398.34107 [2] Cao, J.; Liang, J., Boundedness and stability for Cohen-Grossberg neural network with time-varying delays, Journal of Mathematical Analysis and Applications, 296, 2, 665-685, (2004) · Zbl 1044.92001 [3] Chen, L.; Chai, Y.; Wu, R.; Ma, T.; Zhai, H., Dynamic analysis of a class of fractional-order neural networks with delay, Neurocomputing, 111, 190-194, (2013) [4] Chen, J.; Zeng, Z.; Jiang, P., On the periodic dynamics of memristor-based neural networks with time-varying delays, Information Sciences, 279, 358-373, (2014) · Zbl 1358.34077 [5] Chen, J.; Zeng, Z.; Jiang, P., Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural Networks, 51, 1-8, (2014) · Zbl 1306.34006 [6] Chua, L. O., Memristor-the missing circuit element, IEEE Transactions on Circuit Theory, 18, 5, 507-519, (1971) [7] Cui, W.; Fang, J. A.; Zhang, W.; Wang, X., Finite-time cluster synchronisation of Markovian switching complex networks with stochastic perturbations, IET Control Theory & Applications, 8, 1, 30-41, (2014) · Zbl 1286.93187 [8] Cui, W.; Sun, S.; Fang, J. A.; Xu, Y.; Zhao, L., Finite-time synchronization of Markovian jump complex networks with partially unknown transition rates, Journal of the Franklin Institute, 351, 5, 2543-2561, (2014) · Zbl 1372.93181 [9] De la Sen, M., About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory and Applications, 2011, 1, 1-19, (2011) · Zbl 1219.34102 [10] Ding, D.; Qi, D.; Wang, Q., Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems, IET Control Theory & Applications, 9, 5, 681-690, (2014) [11] Erjaee, G. H.; Momani, S., Phase synchronization in fractional differential chaotic systems, Physics Letters A, 372, 14, 2350-2354, (2008) · Zbl 1220.34004 [12] Filippov, A. F., (Differential equations with discontinuous right-hand sides, Mathematics and its applications, (1988), Kluwer Boston) [13] Gao, Z.; Liao, X., Robust stability criterion of fractional-order functions for interval fractional-order systems, IET Control Theory & Applications, 7, 1, 60-67, (2013) [14] He, Y.; Liu, G. P.; Rees, D.; Wu, M., Stability analysis for neural networks with time-varying interval delay, IEEE Transactions on Neural Networks, 18, 6, 1850-1854, (2007) [15] Hu, C.; Yu, J.; Jiang, H., Finite-time synchronization of delayed neural networks with Cohen-Grossberg type based on delayed feedback control, Neurocomputing, 143, 90-96, (2014) [16] Huang, J.; Li, C.; Huang, T.; He, X., Finite-time lag synchronization of delayed neural networks, Neurocomputing, 139, 145-149, (2014) [17] Itoh, M.; Chua, L. O., Memristor cellular automata and memristor discrete-time cellular neural networks, International Journal of Bifurcation and Chaos, 19, 11, 3605-3656, (2009) · Zbl 1182.37014 [18] Kaslik, E.; Sivasundaram, S., Nonlinear dynamics and chaos in fractional-order neural networks, Neural Networks, 32, 245-256, (2012) · Zbl 1254.34103 [19] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and application of fractional differential equations, (2006), Elsevier New York, NY, USA · Zbl 1092.45003 [20] Li, N.; Cao, J., New synchronization criteria for memristor-based networks: adaptive control and feedback control schemes, Neural Networks, 61, 1-9, (2015) · Zbl 1323.93041 [21] Li, C.; Liao, X.; Yu, J., Synchronization of fractional order chaotic systems, Physical Review E, 68, 6, 1-3, (2003) [22] Liu, X.; Ho, D. W.; Yu, W.; Cao, J., A new switching design to finite-time stabilization of nonlinear systems with applications to neural networks, Neural Networks, 57, 94-102, (2014) · Zbl 1323.93064 [23] Liu, X.; Yu, W.; Cao, J.; Alsaadi, F., Finite-time synchronisation control of complex networks via non-smooth analysis, IET Control Theory & Applications, 9, 8, 1245-1253, (2015) [24] Lu, J. G., Chaotic dynamics of the fractional-order Lu system and its synchronization, Physics Letters A, 354, 4, 305-311, (2006) [25] Lundstrom, B. N.; Higgs, M. H.; Spain, W. J.; Fairhall, A. L., Fractional differentiation by neocortical pyramidal neurons, Nature Neuroscience, 11, 11, 1335-1342, (2008) [26] Milanovic, V.; Zaghloul, M. E., Synchronization of chaotic neural networks and applications to communications, International Journal of Bifurcation and Chaos, 6, 12b, 2571-2585, (1996) · Zbl 1298.94005 [27] Podlubny, I., Fractional differential equations, (1999), Academic Press New York, NY, USA · Zbl 0918.34010 [28] Shen, H.; Park, J. H.; Wu, Z. G., Finite-time synchronization control for uncertain Markov jump neural networks with input constraints, Nonlinear Dynamics, 77, 4, 1709-1720, (2014) · Zbl 1331.92019 [29] Strukov, D. B.; Snider, G. S.; Stewart, D. R.; Williams, R. S., The missing memristor found, Nature, 453, 80-83, (2008) [30] Syed Ali, M., Stability of Markovian jumping recurrent neural networks with discrete and distributed time-varying delays, Neurocomputing, 149, 1280-1285, (2015) [31] Syed Ali, M., Stochastic stability of uncertain recurrent neural networks with Markovian jumping parameters, Acta Mathematica Scientia, 35B, 5, 1122-1136, (2015) · Zbl 1349.93404 [32] Syed Ali, M.; Arik, S.; Saravanakumar, R., Delay-dependent stability criteria of uncertain Markovian jump neural networks with discrete interval and distributed time-varying delays, Neurocomputing, 158, 167-173, (2015) [33] Tan, Z.; Ali, M. K., Associative memory using synchronization in a chaotic neural network, International Journal of Modern Physics C, 12, 01, 19-29, (2001) [34] Tour, J. M.; He, T., The fourth element, Nature, 453, 42-43, (2008) [35] Wang, X.; Chen, Y.; Xi, H.; Li, H.; Dimitrov, D., Spintronic memristor through spin-torque-induced magnetization motion, IEEE Electron Device Letters, 30, 3, 294-297, (2009) [36] Wang, W.; Li, L.; Peng, H.; Xiao, J.; Yang, Y., Synchronization control of memristor-based recurrent neural networks with perturbations, Neural Networks, 53, 8-14, (2014) · Zbl 1307.93038 [37] Wang, H.; Yu, Y.; Wen, G., Stability analysis of fractional-order Hopfield neural networks with time delays, Neural Networks, 55, 98-109, (2014) · Zbl 1322.93089 [38] Wen, S.; Zeng, Z.; Huang, T., Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays, Neurocomputing, 97, 233-240, (2012) [39] Wu, Y.; Cao, J.; Alofi, A.; Abdullah, A. M.; Elaiw, A., Finite-time boundedness and stabilization of uncertain switched neural networks with time-varying delay, Neural Networks, 69, 135-143, (2015) · Zbl 1398.34033 [40] Wu, A.; Zeng, Z., Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays, Neural Networks, 36, 1-10, (2012) · Zbl 1258.34165 [41] Wu, A.; Zeng, Z., Exponential stabilization of memristive neural networks with time delays, IEEE Transactions on Neural Networks and Learning Systems, 23, 12, 1919-1929, (2012) [42] Yan, J.; Li, C., On chaos synchronization of fractional differential equations, Chaos, Solitons & Fractals, 32, 2, 725-735, (2007) · Zbl 1132.37308 [43] Yang, X., Can neural networks with arbitrary delays be finite-timely synchronized?, Neurocomputing, 143, 275-281, (2014) [44] Yang, X.; Cao, J.; Yu, W., Exponential synchronization of memristive Cohen-Grossberg neural networks with mixed delays, Cognitive Neurodynamics, 8, 3, 239-249, (2014) [45] Ye, H.; Gao, J.; Ding, Y., A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 2, 1075-1081, (2007) · Zbl 1120.26003 [46] Yu, J.; Hu, C.; Jiang, H., $$\alpha$$-stability and $$\alpha$$-synchronization for fractional-order neural networks, Neural Networks, 35, 82-87, (2012) · Zbl 1258.34118 [47] Yu, J.; Hu, C.; Jiang, H.; Fan, X., Projective synchronization for fractional neural networks, Neural Networks, 49, 87-95, (2014) · Zbl 1296.34133 [48] Zhang, X.; Feng, G., Global finite-time stabilisation of a class of feedforward non-linear systems, IET Control Theory & Applications, 5, 12, 1450-1457, (2011) [49] Zhang, G.; Shen, Y.; Yin, Q.; Sun, J., Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays, Information Sciences, 232, 386-396, (2013) · Zbl 1293.34094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.