Abstract
This paper presents some theoretical results on dynamical behavior of complex-valued neural networks with discontinuous neuron activations. Firstly, we introduce the Filippov differential inclusions to complex-valued differential equations with discontinuous right-hand side and give the definition of Filippov solution for discontinuous complex-valued neural networks. Secondly, by separating complex-valued neural networks into real and imaginary part, we study the existence of equilibria of the neural networks according to Leray–Schauder alternative theorem of set-valued maps. Thirdly, by constructing appropriate Lyapunov function, we derive the sufficient condition to ensure global asymptotic stability of the equilibria and convergence in finite time. Numerical examples are given to show the effectiveness and merits of the obtained results.
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References
Hirose A (2012) Complex-valued neural networks. Springer, Berlin
Nitta T (2003) Orthogonality of decision boundaries of complex-valued neural networks. Neural Comput 16:73–79
Tanaka G, Aihara K (2009) Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction. IEEE Trans Neural Netw 20:1463–1473
Song R, Xiao W (2014) Adaptive dynamic programming for a class of complex-valued nonlinear systems. IEEE Trans Neural Netw Learn Syst 20(9):1733–1739
Chakravarthy V (2008) Complex-valued neural networks: utilizing high-dimensional parameters. Hershey, New York
Rudin W (1987) Real and complex analysis. Academic, New York
Ozdemir N, Iskender B, Ozgur N (2011) Complex-valued neural network with Mobius activation function. Commun Nonlinear Sci Numer Simul 16:4698–4703
Bohner M, Rao V, Sanyal S (2011) Global stability of complex-valued neural networks on time scales. Differ Equ Dyn Syst 19(1–2):3–11
Yasuaki K, Mitsuo Y (2003) On activation functions for complex-valued neural networks existence of energy functions. Lect Notes Comput Sci 2714:985–992
Liu X, Fang K, Liu B (2009) A synthesis method based on stability analysis for complex-valued Hopfield neural networks. In: Proceedings of the 7th Asian control conference, p 1245–1250
Fang T, Sun J (2013) Stability analysis of complex-valued nonlinear delay differential systems. Syst Control Lett 62:910–914
Gong W, Liang J, Cao J (2015) Matrix measure method for global exponential stability of complex-valued recurrent neural networks with time-varying delays. Neural Netw 70:81–89
Wei J, Zhang C (2004) Stability analysis in a first-order complex differential equations with delay. Nonlinear Anal 59:657–671
Hu J, Wang J (2012) Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans Neural Netw Learn Syst 23(6):853–865
Zhang Z, Lin C, Chen B (2014) Global stability criterion for delayed complex-valued recurrent neural networks. IEEE Trans Neural Netw Learn Syst 25(9):1704–1708
Fang T, Sun J (2014) Further investigation on the stability of complex-valued recurrent neural networks with time delays. IEEE Trans Neural Netw Learn Syst 25(9):1709–1713
Chen X, Song Q (2013) Global stability of complex-valued neural networks with both leakage time delay and discrete time delay on time scales. Neurocomputing 121:254–264
Zou B, Song Q (2013) Boundedness and complete stability of complex-valued neural networks with time delay. IEEE Trans Neural Netw Learn Syst 24(8):1227–1238
Rakkiyappan R, Cao J, Velmurugan G (2014) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 26:84–97
Zhang Z, Yu S (2015) Global asymptotic stability for a class of complex-valued Cohen–Grossberg neural networks with time delay. Neurocomputing 171:1158–1166
Velmurugan G, Cao J (2015) Further analysis of global \(\mu \)-stability of complex-valued neural networks with unbounded time-varying delays. Neural Netw 67:14–27
Hu J, Wang J (2015) Global exponential periodicity and stability of discrete-time complex-valued recurrent neural networks with time-delays. Neural Netw 66:119–130
Pan J, Liu X (2015) Global exponential stability for complex-valued recurrent neural networks with asynchronous time delays. Neurocomputing 164:293–299
Liu X, Chen T (2016) Exponential stability of a class of complex-valued neural networks with time-varying delays. IEEE Trans Neural Netw Learn Syst 27:593–606
Wang H, Huang T, Wang L (2015) Exponential stability of complex-valued memristive recurrent neural networks. IEEE Trans Neural Netw Learn Syst. doi:10.1109/TNNLS.2015.2513001
Hopfield J (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092
Huang Y, Zhang H (2014) Multistability of complex-valued recurrent neural networks with real–imaginary-type activation functions. Appl Math Comput 229:187–200
Rakkiyappan R, Cao J (2014) Multiple \(\mu \)-stability analysis of complex-valued neural networks with unbounded time-varying delays. Neurocomputing 149:594–607
Rakkiyappan R, Sivaranjani K, Velmurugan G (2014) Passivity and passification of memristor based complex-valued recurrent neural networks with interval time varying delays. Neurocomputing 144:391–407
Li X, Rakkiyappan R, Velmurugan G (2014) Dissipativity analysis of memristor based complex-valued neural networks with time-varying delays. Inf Sci 294:645–665
Forti M, Nistri P (2003) Global convergence of neural networks with discontinuous neuron activations. IEEE Trans Circuits Syst I 50(11):1421–1435
Guo Z, Huang L (2009) Generalized Lyapunov method for discontinuous systems. Nonlinear Anal 71(7–8):3083–3092
Forti M, Papini D (2005) Global exponential stability and global convergence in finite time of delayed neural network with infinite gain. IEEE Trans Neural Netw 16:1449–1463
Guo Z, Huang L (2009) LMI conditions for global robust stability of delayed neural networks with discontinuous neuron activations. Appl Math Comput 215:889–900
Guo Z, Huang L (2009) Global output convergence of a class of recurrent delayed neural networks with discontinuous neuron activations. Neural Process Lett 30:213–227
Wang J, Huang L, Guo Z (2009) Dynamical behavior of delayed Hopfield neural networks with discontinuous activations. Appl Math Model 33:1793–1802
Filippov A (1988) Differential equations with discontinuous right-hand side. Kluwer Academic, Boston
Huang L, Guo Z (2009) Global convergence of periodic solution of neural networks with discontinuous activation functions. Chaos Solitons Fractals 42:2351–2356
Wang J, Huang L, Guo Z (2009) Global asymptotic stability of neural networks with discontinuous activations. Neural Netw 22:931–937
Dugundji J, Granas A (2013) Fixed point theory I. Springer, Berlin
Miller R, Michel A (1982) Ordinary differential equations. Academic, Orlando
Acknowledgements
This work was supported by National Natural Science Foundation of China (1573003, 11601143), Natural Science Foundation of Hunan Province of China (13JJ4111, 14JJ3141), a key Project supported by Scientific Research Fund of Hunan Provincial Education Department (15k026, 15A038) and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institution of Hunan Province.
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Wang, Z., Guo, Z., Huang, L. et al. Dynamical Behavior of Complex-Valued Hopfield Neural Networks with Discontinuous Activation Functions. Neural Process Lett 45, 1039–1061 (2017). https://doi.org/10.1007/s11063-016-9563-5
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DOI: https://doi.org/10.1007/s11063-016-9563-5