On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network
Abstract
In this work we report numerical results involving a certain Hopfield-type three-neurons network, with the hyperbolic tangent as the activation function. Specifically, we investigate a place of a two-dimensional parameter-space of
this system where typical periodic structures, the so-called shrimps, are embedded in a chaotic region. We show that these structures are organized themselves as a spiral that coil up toward a focal point, while undergo period-adding bifurcations. We also indicate the locations along this spiral in the parameter-space, where such bifurcations happen.
Keywords
Chaos , Lyapunov exponents spectrum , Parameter space , Period-adding bifurcation
References
- [1] J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA, 81 (1984), 3088–3092.
- [2] E. Korner, R. Kupper, M. K. M. Rahman, Y. Shkuro, Neurocomputing Research Developments, Nova Science Publishers, New York, 2007.
- [3] W. Z. Huang, Y. Huang, Chaos, bifurcations and robustness of a class of Hopfield neural networks, Int. J. Bifurcation and Chaos, 21 (2011), 885–895.
- [4] P. F. Chen, Z. Q. Chen, and W. J. Wu, A novel chaotic system with one source and two saddle-foci in Hopfield neural networks, Chin. Phys. B, 19 (2010), 040509.
- [5] P. Zheng, W. Tang, J. Hang, Some novel double-scroll chaotic attractors in Hopfield networks, Neurocomputing, 73 (2010), 2280–2285.
- [6] A. C. Mathias and P. C. Rech, Hopfield neural network: The hyperbolic tangent and the piecewise-linear activation functions, Neural Networks, 34 (2012), 42–45.
- [7] P. C. Rech, Period-adding and spiral organization of the periodicity in a Hopfield neural network, Int. J. Mach. Learn. & Cyber., 6 (2015), 1–6.
- [8] A. Wolf , J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285–317.
- [9] C. Bonatto, J. A. C. Gallas, Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit, Phys. Rev. Lett., 101 (2008), 054101.
- [10] J. A. C. Gallas, The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows, Int. J. Bifurcation and Chaos, 20 (2010), 197–211.
