Stability and Hopf bifurcation analysis of a complex-valued Wilson–Cowan neural network with time delay

Abstract Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results.

[1]  Jian Xu,et al.  Bursting Near Bautin bifurcation in a Neural Network with Delay Coupling , 2009, Int. J. Neural Syst..

[2]  Jinde Cao,et al.  Bifurcations in a delayed fractional complex-valued neural network , 2017, Appl. Math. Comput..

[3]  Bo Zhou,et al.  Boundedness and complete stability of complex-valued neural networks with time delay. , 2013, IEEE transactions on neural networks and learning systems.

[4]  Marcos Eduardo Valle,et al.  An introduction to complex-valued recurrent correlation neural networks , 2014, 2014 International Joint Conference on Neural Networks (IJCNN).

[5]  Jian Xu,et al.  Bifurcation and Chaos Analysis for a delayed Two-Neural Network with a Variation Slope Ratio in the Activation Function , 2012, Int. J. Bifurc. Chaos.

[6]  Akira Hirose.  Complex-Valued Neural Networks: Distinctive Features , 2012 .

[7]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[8]  Xiaofeng Liao,et al.  Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay , 2008 .

[9]  Tohru Nitta,et al.  Solving the XOR problem and the detection of symmetry using a single complex-valued neuron , 2003, Neural Networks.

[10]  X. Liao,et al.  Hopf bifurcation in love dynamical models with nonlinear couples and time delays , 2007 .

[11]  Jian Xu,et al.  Codimension-two bursting analysis in the delayed neural system with external stimulations , 2012 .

[12]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[13]  Tohru Nitta,et al.  Local minima in hierarchical structures of complex-valued neural networks , 2013, Neural Networks.

[14]  Tianping Chen,et al.  Global Exponential Stability for Complex-Valued Recurrent Neural Networks With Asynchronous Time Delays , 2015, IEEE Transactions on Neural Networks and Learning Systems.

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

[16]  Xiaofeng Liao,et al.  Stability and Hopf bifurcation of a complex-valued neural network with two time delays , 2015 .

[17]  Dong Xie,et al.  Global exponential stability of periodic solution for delayed complex-valued neural networks with impulses , 2016, Neurocomputing.

[18]  Bo Du,et al.  Global Exponential Stability of Periodic Solution for Neutral-Type Complex-Valued Neural Networks , 2016 .

[19]  X. Liao,et al.  Dynamics of the congestion control model in underwater wireless sensor networks with time delay , 2016 .

[20]  Tao Dong,et al.  Hopf bifurcation in a love-triangle model with time delays , 2017, Neurocomputing.

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

[22]  Jun Wang,et al.  Global exponential periodicity and stability of discrete-time complex-valued recurrent neural networks with time-delays , 2015, Neural Networks.