Global Stability, Bifurcation, and Chaos Control in a Delayed Neural Network Model

Conditions for the global asymptotic stability of delayed artificial neural network model of n (≥3) neurons have been derived. For bifurcation analysis with respect to delay we have considered the model with three neurons and used suitable transformation on multiple time delays to reduce it to a system with single delay. Bifurcation analysis is discussed with respect to single delay. Numerical simulations are presented to verify the analytical results. Using numerical simulation, the role of delay and neuronal gain parameter in changing the dynamics of the neural network model has been discussed.

[1]  Qu,et al.  Controlling spatiotemporal chaos in coupled map lattice systems. , 1994, Physical review letters.

[2]  E. Hunt.  Stabilizing high-period orbits in a chaotic system: The diode resonator. , 1991 .

[3]  K. Aihara,et al.  Chaotic neural networks , 1990 .

[4]  P. Das,et al.  Bifurcation and Chaos in Delayed Cellular Neural Network Model , 2014 .

[5]  W. Freeman.  Simulation of chaotic EEG patterns with a dynamic model of the olfactory system , 1987, Biological Cybernetics.

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

[7]  Kazuyuki Aihara,et al.  Global searching ability of chaotic neural networks , 1999 .

[8]  A. Das,et al.  Chaos in a Three-Dimensional General Model of Neural Network , 2002, Int. J. Bifurc. Chaos.

[9]  A. Babloyantz,et al.  Low-dimensional chaos in an instance of epilepsy. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Ping Zhu,et al.  Controlling chaos in a chaotic neural network , 2003, Neural Networks.

[11]  K. Aihara,et al.  Chaos and phase locking in normal squid axons , 1987 .

[12]  Jonathan D. Cohen,et al.  The effects of neural gain on attention and learning , 2013, Nature Neuroscience.

[13]  Edward Ott,et al.  Controlling chaos , 2006, Scholarpedia.

[14]  Junjie Wei,et al.  Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays , 2005 .

[15]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[16]  Jürgen Jost,et al.  Delays, connection topology, and synchronization of coupled chaotic maps. , 2004, Physical review letters.

[17]  Kaplan,et al.  Subthreshold dynamics in periodically stimulated squid giant axons. , 1996, Physical review letters.

[18]  Kazuyuki Aihara,et al.  Associative Dynamics in a Chaotic Neural Network , 1997, Neural Networks.

[19]  Martin Bohner,et al.  Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback , 2010, Math. Comput. Model..

[20]  K. Aihara,et al.  12. Chaotic oscillations and bifurcations in squid giant axons , 1986 .

[21]  Pritha Das,et al.  Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays , 2013 .

[22]  Jinde Cao,et al.  Bifurcation and control in a neural network with small and large delays , 2013, Neural Networks.

[23]  Robert C. Hilborn,et al.  Chaos and Nonlinear Dynamics , 2000 .