Research Article

Year 2022,
Volume: 5 Issue: 2, 92 - 98, 30.06.2022
### Abstract

### Supporting Institution

### Project Number

### References

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.

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.

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

0

- [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
### Abstract

### Project Number

### References

0

- [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.

Primary Language | English |
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Subjects | Mathematical Sciences |

Journal Section | Articles |

Authors | |

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 |

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