Chaos and hyperchaos in a Hopfield neural network

In this paper we investigate numerically the parameter-space of an autonomous system of four nonlinear first-order ordinary differential equations, which represents a Hopfield neural network with four neurons. The study considers three independent two-dimensional cross-sections of the three-dimensional parameter-space generated by this mathematical model, every constructed considering Lyapunov exponent values. We show that is possible to completely characterize the dynamics of the system based in these three plots, which are representative of the three-dimensional parameter-space as a whole.

[1]  Paulo C. Rech,et al.  Self-similar structures in a 2D parameter-space of an inductorless Chua's circuit , 2008 .

[2]  Holokx A. Albuquerque,et al.  Complex periodic structures in bi-dimensional bifurcation diagrams of a RLC circuit model with a nonlinear NDC device , 2009 .

[3]  Paulo C. Rech,et al.  Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity. , 2010, Chaos.

[4]  Tommy W. S. Chow,et al.  Neurocomputing Research Developments , 2007 .

[5]  J. Gallas,et al.  Accumulation horizons and period adding in optically injected semiconductor lasers. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Yan Huang,et al.  Chaos and Bifurcation in a New Class of Simple Hopfield Neural Network , 2006, ISNN.

[7]  Marco Storace,et al.  Experimental bifurcation diagram of a circuit-implemented neuron model , 2010 .

[8]  J. G. Freire,et al.  Relative abundance and structure of chaotic behavior: the nonpolynomial Belousov-Zhabotinsky reaction kinetics. , 2009, The Journal of chemical physics.

[9]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Xiao-Song Yang,et al.  Hyperchaos in Hopfield-type neural networks , 2005, Neurocomputing.

[11]  J. Gallas,et al.  Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit. , 2008, Physical review letters.

[12]  Paulo C. Rech,et al.  Dynamics of a neuron model in different two-dimensional parameter-spaces , 2011 .

[13]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .