Finite-time stability of fractional-order BAM neural networks with distributed delay. (English) Zbl 1474.93177

Summary: Based on the theory of fractional calculus, the generalized Gronwall inequality and estimates of mittag-Leffer functions, the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay is investigated in this paper. An illustrative example is also given to demonstrate the effectiveness of the obtained result.


93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34K20 Stability theory of functional-differential equations
34K37 Functional-differential equations with fractional derivatives
92B20 Neural networks for/in biological studies, artificial life and related topics
93A14 Decentralized systems
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[1] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[2] Podlubny, I., Fractional Differential Equations (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[3] Soczkiewicz, E., Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23, 397-404 (2002)
[4] Kulish, V. V.; Lage, J. L., Application of fractional calculus to fluid mechanics, Journal of Fluids Engineering, 124, 3, 803-806 (2002)
[5] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1092.45003
[6] Sabatier, J.; Agrawal, O. P.; Machado, J., Theoretical Developments and Applications, Advance in Fractional Calculus (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1116.00014
[7] Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Bifurcation and chaos in noninteger order cellular neural networks, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 8, 7, 1527-1539 (1998) · Zbl 0936.92006
[8] Arena, P.; Fortuna, L.; Porto, D., Chaotic behavior in noninteger-order cellular neural networks, Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 61, 1, 776-781 (2000)
[9] Lazarević, M. P., Finite time stability analysis of \(\operatorname{ PD }^\alpha\) fractional control of robotic time-delay systems, Mechanics Research Communications, 33, 2, 269-279 (2006) · Zbl 1192.70008
[10] Boroomand, A.; Menhaj, M., Fractional-order Hopfield neural networks, Proceedings of the Natural Computation International Conference
[11] Kaslik, E.; Sivasundaram, S., Dynamics of fractional-order neural networks, Proceedings of the International Joint Conference on Neural Network (IJCNN ’11)
[12] Kaslik, E.; Sivasundaram, S., Nonlinear dynamics and chaos in fractional-order neural networks, Neural Networks, 32, 245-256 (2012) · Zbl 1254.34103
[13] Zhang, R.; Qi, D.; Wang, Y., Dynamics analysis of fractional order three-dimensional Hopfield neural network, Proceedings of the 6th International Conference on Natural Computation (ICNC ’10)
[14] Delavari, H.; Baleanu, D.; Sadati, J., Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dynamics, 67, 4, 2433-2439 (2012) · Zbl 1243.93081
[15] Li, Y.; Chen, Y.; Podlubny, I., Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 8, 1965-1969 (2009) · Zbl 1185.93062
[16] Li, Y.; Chen, Y.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: lyapunov direct method and generalized Mittag-Leffler stability, Computers & Mathematics with Applications, 59, 5, 1810-1821 (2010) · Zbl 1189.34015
[17] Gu, K.; Kharitonov, V. L.; Chen, J., Stability of Time-delay Systems (2003), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1039.34067
[18] Arik, S., An analysis of global asymptotic stability of delayed cellular neural networks, IEEE Transactions on Neural Networks, 13, 5, 1239-1242 (2002)
[19] Tian, J.; Xiong, W.; Xu, F., Improved delay-partitioning method to stability analysis for neural networks with discrete and distributed time-varying delays, Applied Mathematics and Computation, 233, 152-164 (2014) · Zbl 1334.92025
[20] Tian, J.; Li, Y.; Zhao, J.; Zhong, S., Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates, Applied Mathematics and Computation, 218, 9, 5769-5781 (2012) · Zbl 1248.34123
[21] Chen, L.; Chai, Y.; Wu, R.; Zhai, T., Dynamic analysis of a class of fractional-order neural networks with delay, Neurocomputing, 111, 190-194 (2013)
[22] Wu, R.-C.; Hei, X.-D.; Chen, L.-P., Finite-time stability of fractional-order neural networks with delay, Communications in Theoretical Physics, 60, 2, 189-193 (2013) · Zbl 1284.92016
[23] Alofi, A.; Cao, J.; Elaiw, A.; Al-Mazrooei, A., Delay-dependent stability criterion of caputo fractional neural networks with distributed delay, Discrete Dynamics in Nature and Society, 2014 (2014) · Zbl 1418.92005
[24] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences of the United States of America, 81, 10, 3088-3092 (1984) · Zbl 1371.92015
[25] Kosko, B., Bidirectional associative memories, IEEE Transactions on Systems, Man, and Cybernetics, 18, 1, 49-60 (1988)
[26] Arik, S.; Tavsanoglu, V., Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays, Neurocomputing, 68, 1-4, 161-176 (2005)
[27] Senan, S.; Arik, S.; Liu, D., New robust stability results for bidirectional associative memory neural networks with multiple time delays, Applied Mathematics and Computation, 218, 23, 11472-11482 (2012) · Zbl 1277.93066
[28] Liu, B., Global exponential stability for BAM neural networks with time-varying delays in the leakage terms, Nonlinear Analysis: Real World Applications, 14, 1, 559-566 (2013) · Zbl 1260.34138
[29] Raja, R.; Anthoni, S. M., Global exponential stability of BAM neural networks with time-varying delays: the discrete-time case, Communications in Nonlinear Science and Numerical Simulation, 16, 2, 613-622 (2011) · Zbl 1221.39025
[30] 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 (2011) · Zbl 1219.34102
[31] 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
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