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Multistability of complex-valued neural networks with discontinuous activation functions. (English) Zbl 1432.34104

Summary: In this paper, based on the geometrical properties of the discontinuous activation functions and the Brouwer’s fixed point theory, the multistability issue is tackled for the complex-valued neural networks with discontinuous activation functions and time-varying delays. To address the network with discontinuous functions, Filippov solution of the system is defined. Through rigorous analysis, several sufficient criteria are obtained to assure the existence of \(25^n\) equilibrium points. Among them, \(9^n\) points are locally stable and \(16^n-9^n\) equilibrium points are unstable. Furthermore, to enlarge the attraction basins of the \(9^n\) equilibrium points, some mild conditions are imposed. Finally, one numerical example is provided to illustrate the effectiveness of the obtained results.

MSC:

34K39 Discontinuous functional-differential equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34K21 Stationary solutions of functional-differential equations
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[1] Amin, M. F.; Murase, K., Single-layered complex-valued neural network for real-valued classification problems, Neurocomputing, 72, 4-6, 945-955 (2009)
[2] Bao, G.; Zeng, Z., Multistability of periodic delayed recurrent neural network with memristors, Neural Computing & Applications, 23, 7, 1963-1967 (2013)
[3] Cha, I.; Kassam, S. A., Channel equalization using adaptive complex radial basis function networks, IEEE Journal on Selected Areas in Communications, 13, 1, 122-131 (1995)
[4] Chakravarthy, S. V.; Ghosh, J., A complex-valued associative memory for storing patterns as oscillatory states, Biological Cybernetics, 75, 3, 229-238 (1996) · Zbl 0864.92003
[5] Cheng, C.-Y.; Lin, K.-H.; Shih, C.-W., Multistability in recurrent neural networks, SIAM Journal on Applied Mathematics, 66, 4, 1301-1320 (2006) · Zbl 1106.34048
[6] Fang, T.; Sun, J., Further investigate the stability of complex-valued recurrent neural networks with time-delays, IEEE Transactions on Neural Networks and Learning Systems, 25, 9, 1709-1713 (2014)
[7] Filippov, A. F., Differential equations with discontinuous right-hand sides (1988), Kluwer: Kluwer Dordrecht · Zbl 0664.34001
[8] Goh, S. L.; Mandic, D. P., A complex-valued RTRL algorithm for recurrent neural networks, Neural Computation, 16, 12, 2699-2713 (2004) · Zbl 1062.68098
[9] Goh, S. L.; Mandic, D. P., An augmented extended Kalman filter algorithm for complex-valued recurrent neural networks, Neural Computation, 19, 4, 1039-1055 (2007) · Zbl 1118.68118
[10] Hirose, A., Dynamics of fully complex-valued neural networks, Electronics Letters, 28, 16, 1492-1494 (1992)
[11] Hirose, A., Complex-Valued Neural Networks (2006), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 1126.68067
[12] Hu, J.; Wang, J., Global stability of complex-valued recurrent neural networks with time-delays, IEEE Transactions on Neural Networks and Learning Systems, 23, 6, 853-865 (2012)
[14] Huang, G.; Cao, J., Multistability of neural networks with discontinuous activation function, Communications in Nonlinear Science and Numerical Simulation, 13, 10, 2279-2289 (2008) · Zbl 1221.34131
[15] Huang, G.; Cao, J., Delay-dependent multistability in recurrent neural networks, Neural Networks, 23, 2, 201-209 (2010) · Zbl 1400.34115
[16] Huang, Y.; Zhang, H.; Wang, Z., Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions, Neurocomputing, 91, 21-28 (2012)
[17] Huang, Y.; Zhang, H.; Wang, Z., Multistability of complex-valued recurrent neural networks with real-imaginary-type activation functions, Applied Mathematics and Computation, 229, 187-200 (2014) · Zbl 1364.92003
[18] Jankowski, S.; Lozowski, A.; Zurada, J. M., Complex-valued multistate neural associative memory, IEEE Transactions on Neural Networks, 7, 6, 1491-1496 (1996)
[19] Khajanchi, S.; Banerjee, S., Stability and bifurcation analysis of delay induced tumor immune interaction model, Applied Mathematics and Computation, 248, 652-671 (2014) · Zbl 1338.92048
[20] Kim, T.; Adali, T., Fully complex multi-layer perceptron network for nonlinear signal processing, Journal of VLSI Signal Processing, 32, 29-43 (2002) · Zbl 1016.68070
[21] Kim, T.; Adali, T., Approximation by fully complex multilayer perceptrons, Neural Computation, 15, 7, 1641-1666 (2003) · Zbl 1085.68631
[22] Lee, D. L., Relaxation of the stability condition of the complex-valued neural networks, IEEE Transactions on Neural Networks, 12, 5, 1260-1262 (2001)
[23] Lee, D.-L., Improvements of complex-valued Hopfield associative memory by using generalized projection rules, IEEE Transactions on Neural Networks, 17, 5, 1341-1347 (2006)
[24] Li, C.; Liao, X.; Yu, J., Complex-valued recurrent neural network with IIR neuron model: training and applications, Circuits, Systems and Signal Processing, 21, 5, 461-471 (2002) · Zbl 1057.68086
[25] Liang, J.; Wang, Z.; Liu, Y.; Liu, X., State estimation for two-dimensional complex networks with randomly occuring onlinearities and randomly varying sensor delays, International Journal of Robust and Nonlinear Control, 24, 1, 18-38 (2014) · Zbl 1417.93304
[26] Mostafa, M.; Teich, W. G.; Lindner, J., Local stability analysis of discrete-time, continuous-state, complex-valued recurrent neural networks with inner state feedback, IEEE Transactions on Neural Networks and Learning Systems, 25, 4, 830-836 (2014)
[27] Nie, X.; Zheng, W. X., Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays, Neural Networks, 65, 65-79 (2015) · Zbl 1394.68305
[28] Nie, X.; Zheng, W. X., Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Transactions on Cybernetics, 46, 3, 679-693 (2016)
[29] Nitta, T., Orthogonality of decision boundaries in complex-valued neural networks, Neural Computation, 16, 1, 73-97 (2004) · Zbl 1084.68105
[30] Rakkiyappan, R.; Sivaranjani, R.; Velmurugana, G.; Cao, J., Analysis of global \(O(t^{- \alpha})\) stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays, Neural Networks, 77, 51-69 (2016) · Zbl 1417.34194
[31] Rakkiyappan, R.; Velmurugan, G.; Cao, J., Multiple \(\mu \)-stability analysis of complex-valued neural networks with unbounded time-varying delays, Neurocomputing, 149, 594-607 (2015), Part B · Zbl 1398.34103
[32] Rao, V. S.H.; Murthy, G. R., Global dynamics of a class of complex valued neural networks, International Journal of Neural Systems, 18, 2, 165-171 (2008)
[33] Savitha, R.; Suresh, S.; Sundararajan, N., A fully complex-valued radial basis function network and its learning algorithm, International Journal of Neural Systems, 19, 4, 253-267 (2009)
[34] Savitha, R.; Suresh, S.; Sundararajan, N., Projection-based fast learning fully complex-valued relaxation neural network, IEEE Transactions on Neural Networks and Learning Systems, 24, 4, 529-541 (2013)
[35] Song, Q.; Zhao, Z.; Liu, Y., Stability analysis of complex-valued neural networks with probabilistic time-varying delays, Neurocomputing, 159, 96-104 (2015)
[36] Tanaka, G.; Aihara, K., Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction, IEEE Transactions on Neural Networks, 20, 9, 1463-1473 (2009)
[37] Wang, L.; Chen, T., Multistability of neural networks with mexican-hat-type activation functions, IEEE Transactions on Neural Networks and Learning Systems, 23, 11 (2012)
[38] Wang, L.; Chen, T., Multiple \(\mu \)-stability of neural networks with unbounded time-varying delays, Neural Networks, 53, 109-118 (2014) · Zbl 1307.93365
[39] Wang, Z.; Ding, S.; Huang, Z.; Zhang, H., Exponential stability and stabilization of delayed memristive neural networks based on quadratic convex combination method, IEEE Transactions on Neural Networks and Learning Systems (2015)
[40] Wang, Y.; Huang, L., Global stability analysis of competitive neural networks with mixed time-varying delays and discontinuous neuron activations, Neurocomputing, 152, 85-96 (2015)
[41] Wang, Z.; Liu, L.; Shan, Q.-H.; Zhang, H., Stability criteria for recurrent neural networks with time-varying delay based on secondary delay partitioning method, IEEE Transactions on Neural Networks and Learning Systems, 26, 10, 2589-2595 (2015)
[42] Wang, L.; Lu, W.; Chen, T., Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions, Neural Networks, 23, 2, 189-200 (2010) · Zbl 1409.34025
[43] Xu, X.; Zhang, J.; Shi, J., Exponential stability of complex-valued neural networks with mixed delays, Neurocomputing, 128, 483-490 (2014)
[44] Zeng, X.; Li, C.; Huang, T.; He, X., Stability analysis of complex-valued impulsive systems with time delay, Applied Mathematics and Computation, 256, 75-82 (2015) · Zbl 1338.34132
[45] Zhang, Z.; Lin, C.; Chen, B., Global stability criterion for delayed complex-valued recurrent neural networks, IEEE Transactions on Neural Networks and Learning Systems, 25, 9, 1704-1708 (2014)
[46] Zhou, B.; Song, Q., Boundedness and complete stability of complex-valued neural networks with time delay, IEEE Transactions on Neural Networks and Learning Systems, 24, 8, 1227-1238 (2013)
[47] Zhou, W.; Zurada, J. M., Discrete-time recurrent neural networks with complex-valued linear threshold neurons, IEEE Transactions on Circuits and Systems-II: Express Briefs, 56, 8, 669-673 (2009)
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