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On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network

Year 2022, Volume: 5 Issue: 2, 92 - 98, 30.06.2022
https://doi.org/10.33434/cams.1064713

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.

Supporting Institution

Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico-CNPq, and Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina-FAPESC, Brazilian Agencies.

Project Number

0

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.
  • [11] H. A. Albuquerque, P. C. Rech, Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit, Int. J. Circ. Theor. Appl., 40 (2012), 189–194.
  • [12] R. Barrio, F. Blesa, S. Serrano, Qualitative analysis of the R¨ossler equations: Bifurcations of limit cycles and chaotic attractors, Physica D, 238 (2009), 1087–1100.
  • [13] X. F. Li, Y. T. L. Andrew, Y. D. Chu, Symmetry and period-adding windows in a modified optical injection semiconductor laser model, Chin. Phys. Lett., 29 (2012), 010201.
  • [14] C. Stegemann, P. C. Rech, Organization of the dynamics in a parameter plane of a tumor growth mathematical model, Int. J. Bifurcation and Chaos, 24 (2014), 1450023.
  • [15] R. A. da Silva, P. C. Rech, Spiral periodic structures in a parameter plane of an ecological model, Appl. Math. Comput., 254 (2015), 9–13.
  • [16] P. C. Rech, Spiral organization of periodic structures in the Lorenz-Stenflo system, Phys. Scr., 91 (2016), 075201.
  • [17] A. da Silva, P. C. Rech, Numerical investigation concerning the dynamics in parameter planes of the Ehrhard-M¨uller System, Chaos Solitons Fractals, 110 (2018), 152–157.
  • [18] R. Barrio, F. Blesa, S. Serrano, A. Shilnikov, Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci, Phys. Rev. E, 84 (2011), 035201.
  • [19] R. Vitolo, P. Glendinning, J. A. C. Gallas, Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows, Phys. Rev. E, 84 (2011), 016216.
  • [20] R. Stoop, P. Benner, Y. Uwate, Real-world existence and origins of the spiral organization of shrimp-shaped domains, Phys. Rev. Lett., 105 (2010), 074102.
Year 2022, Volume: 5 Issue: 2, 92 - 98, 30.06.2022
https://doi.org/10.33434/cams.1064713

Abstract

Project Number

0

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.
  • [11] H. A. Albuquerque, P. C. Rech, Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit, Int. J. Circ. Theor. Appl., 40 (2012), 189–194.
  • [12] R. Barrio, F. Blesa, S. Serrano, Qualitative analysis of the R¨ossler equations: Bifurcations of limit cycles and chaotic attractors, Physica D, 238 (2009), 1087–1100.
  • [13] X. F. Li, Y. T. L. Andrew, Y. D. Chu, Symmetry and period-adding windows in a modified optical injection semiconductor laser model, Chin. Phys. Lett., 29 (2012), 010201.
  • [14] C. Stegemann, P. C. Rech, Organization of the dynamics in a parameter plane of a tumor growth mathematical model, Int. J. Bifurcation and Chaos, 24 (2014), 1450023.
  • [15] R. A. da Silva, P. C. Rech, Spiral periodic structures in a parameter plane of an ecological model, Appl. Math. Comput., 254 (2015), 9–13.
  • [16] P. C. Rech, Spiral organization of periodic structures in the Lorenz-Stenflo system, Phys. Scr., 91 (2016), 075201.
  • [17] A. da Silva, P. C. Rech, Numerical investigation concerning the dynamics in parameter planes of the Ehrhard-M¨uller System, Chaos Solitons Fractals, 110 (2018), 152–157.
  • [18] R. Barrio, F. Blesa, S. Serrano, A. Shilnikov, Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci, Phys. Rev. E, 84 (2011), 035201.
  • [19] R. Vitolo, P. Glendinning, J. A. C. Gallas, Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows, Phys. Rev. E, 84 (2011), 016216.
  • [20] R. Stoop, P. Benner, Y. Uwate, Real-world existence and origins of the spiral organization of shrimp-shaped domains, Phys. Rev. Lett., 105 (2010), 074102.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Angela Da Silva This is me

Paulo Rech

Project Number 0
Publication Date June 30, 2022
Submission Date January 31, 2022
Acceptance Date June 27, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Da Silva, A., & Rech, P. (2022). On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network. Communications in Advanced Mathematical Sciences, 5(2), 92-98. https://doi.org/10.33434/cams.1064713
AMA Da Silva A, Rech P. On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network. Communications in Advanced Mathematical Sciences. June 2022;5(2):92-98. doi:10.33434/cams.1064713
Chicago Da Silva, Angela, and Paulo Rech. “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”. Communications in Advanced Mathematical Sciences 5, no. 2 (June 2022): 92-98. https://doi.org/10.33434/cams.1064713.
EndNote Da Silva A, Rech P (June 1, 2022) On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network. Communications in Advanced Mathematical Sciences 5 2 92–98.
IEEE A. Da Silva and P. Rech, “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”, Communications in Advanced Mathematical Sciences, vol. 5, no. 2, pp. 92–98, 2022, doi: 10.33434/cams.1064713.
ISNAD Da Silva, Angela - Rech, Paulo. “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”. Communications in Advanced Mathematical Sciences 5/2 (June 2022), 92-98. https://doi.org/10.33434/cams.1064713.
JAMA Da Silva A, Rech P. On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network. Communications in Advanced Mathematical Sciences. 2022;5:92–98.
MLA Da Silva, Angela and Paulo Rech. “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”. Communications in Advanced Mathematical Sciences, vol. 5, no. 2, 2022, pp. 92-98, doi:10.33434/cams.1064713.
Vancouver Da Silva A, Rech P. On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network. Communications in Advanced Mathematical Sciences. 2022;5(2):92-8.

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