Multistability of complex-valued neural networks with discontinuous activation functions

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 25n equilibrium points. Among them, 9n points are locally stable and 16n-9n equilibrium points are unstable. Furthermore, to enlarge the attraction basins of the 9n equilibrium points, some mild conditions are imposed. Finally, one numerical example is provided to illustrate the effectiveness of the obtained results.

[1]  Garimella Rama Murthy,et al.  Global Dynamics of a Class of Complex Valued Neural Networks , 2008, Int. J. Neural Syst..

[2]  Jinde Cao,et al.  Delay-dependent multistability in recurrent neural networks , 2010, Neural Networks.

[3]  Jacek M. Zurada,et al.  Discrete-Time Recurrent Neural Networks With Complex-Valued Linear Threshold Neurons , 2009, IEEE Trans. Circuits Syst. II Express Briefs.

[4]  Juebang Yu,et al.  Complex-Valued Recurrent Neural Network with IIR Neuron Model: Training and Applications , 2002 .

[5]  Jinde Cao,et al.  Multiple μ-stability analysis of complex-valued neural networks with unbounded time-varying delays , 2015, Neurocomputing.

[6]  Jun Wang,et al.  Global Stability of Complex-Valued Recurrent Neural Networks With Time-Delays , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[7]  Saleem A. Kassam,et al.  Channel Equalization Using Adaptive Complex Radial Basis Function Networks , 1995, IEEE J. Sel. Areas Commun..

[8]  Danilo P. Mandic,et al.  A Complex-Valued RTRL Algorithm for Recurrent Neural Networks , 2004, Neural Computation.

[9]  Bing Chen,et al.  Global Stability Criterion for Delayed Complex-Valued Recurrent Neural Networks , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[10]  Tianping Chen,et al.  Multiple µ-stability of neural networks with unbounded time-varying delays , 2014, Neural Networks.

[11]  Jun Wang,et al.  Multistability and Multiperiodicity Analysis of Complex-Valued Neural Networks , 2014, ISNN.

[12]  Jinde Cao,et al.  Analysis of global O(t-α) stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays , 2016, Neural Networks.

[13]  Tianping Chen,et al.  Multistability of Neural Networks With Mexican-Hat-Type Activation Functions , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[14]  Huaguang Zhang,et al.  Stability Criteria for Recurrent Neural Networks With Time-Varying Delay Based on Secondary Delay Partitioning Method , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[15]  Jinde Cao,et al.  Multistability of neural networks with discontinuous activation function , 2008 .

[16]  Wei Xing Zheng,et al.  Dynamical Behaviors of Multiple Equilibria in Competitive Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions , 2016, IEEE Transactions on Cybernetics.

[17]  Narasimhan Sundararajan,et al.  Projection-Based Fast Learning Fully Complex-Valued Relaxation Neural Network , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[18]  Tülay Adali,et al.  Approximation by Fully Complex Multilayer Perceptrons , 2003, Neural Computation.

[19]  D.-L. Lee,et al.  Improvements of Complex-Valued Hopfield Associative Memory by Using Generalized Projection Rules , 2006, IEEE Transactions on Neural Networks.

[20]  Donq-Liang Lee,et al.  Relaxation of the stability condition of the complex-valued neural networks , 2001, IEEE Trans. Neural Networks.

[21]  Jacek M. Zurada,et al.  Complex-valued multistate neural associative memory , 1996, IEEE Trans. Neural Networks.

[22]  S. L. Goh,et al.  An Augmented Extended Kalman Filter Algorithm for Complex-Valued Recurrent Neural Networks , 2007, Neural Computation.

[23]  Jitao Sun,et al.  Further Investigate the Stability of Complex-Valued Recurrent Neural Networks With Time-Delays , 2014 .

[24]  Sandip Banerjee,et al.  Stability and bifurcation analysis of delay induced tumor immune interaction model , 2014, Appl. Math. Comput..

[25]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[26]  Zhigang Zeng,et al.  Multistability of periodic delayed recurrent neural network with memristors , 2012, Neural Computing and Applications.

[27]  Wei Xing Zheng,et al.  Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays , 2015, Neural Networks.

[28]  Akira Hirose,et al.  Complex-Valued Neural Networks , 2006, Studies in Computational Intelligence.

[29]  Tohru Nitta,et al.  Orthogonality of Decision Boundaries in Complex-Valued Neural Networks , 2004, Neural Computation.

[30]  Joydeep Ghosh,et al.  A complex-valued associative memory for storing patterns as oscillatory states , 1996, Biological Cybernetics.

[31]  Huaguang Zhang,et al.  Multistability of complex-valued recurrent neural networks with real-imaginary-type activation functions , 2014, Appl. Math. Comput..

[32]  Huaguang Zhang,et al.  Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions , 2012, Neurocomputing.

[33]  Qiankun Song,et al.  Boundedness and Complete Stability of Complex-Valued Neural Networks With Time Delay , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[34]  Jürgen Lindner,et al.  Local Stability Analysis of Discrete-Time, Continuous-State, Complex-Valued Recurrent Neural Networks With Inner State Feedback , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[35]  Kazuyuki Murase,et al.  Single-layered complex-valued neural network for real-valued classification problems , 2009, Neurocomputing.

[36]  N. Sundararajan,et al.  A fully complex-valued radial basis function network and its learning algorithm. , 2009 .

[37]  Tianping Chen,et al.  Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions , 2010, Neural Networks.

[38]  CHIH-WEN SHIH,et al.  Multistability in Recurrent Neural Networks , 2006, SIAM J. Appl. Math..

[39]  Tülay Adali,et al.  Fully Complex Multi-Layer Perceptron Network for Nonlinear Signal Processing , 2002, J. VLSI Signal Process..

[40]  Zidong Wang,et al.  State estimation for two‐dimensional complex networks with randomly occurring nonlinearities and randomly varying sensor delays , 2014 .

[41]  Akira Hirose,et al.  Dynamics of fully complex-valued neural networks , 1992 .

[42]  Chuandong Li,et al.  Stability analysis of complex-valued impulsive systems with time delay , 2015, Appl. Math. Comput..

[43]  Zhenjiang Zhao,et al.  Stability analysis of complex-valued neural networks with probabilistic time-varying delays , 2015, Neurocomputing.

[44]  Xiaohui Xu,et al.  Exponential stability of complex-valued neural networks with mixed delays , 2014, Neurocomputing.

[45]  Lihong Huang,et al.  Global stability analysis of competitive neural networks with mixed time-varying delays and discontinuous neuron activations , 2015, Neurocomputing.

[46]  Gouhei Tanaka,et al.  Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction , 2009, 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence).

[47]  Huaguang Zhang,et al.  Exponential Stability and Stabilization of Delayed Memristive Neural Networks Based on Quadratic Convex Combination Method , 2016, IEEE Transactions on Neural Networks and Learning Systems.