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

Angela Da Silva; Paulo Rech

doi:10.33434/cams.1064713

Araştırma Makalesi

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

Yıl 2022, , 92 - 98, 30.06.2022

Angela Da Silva Paulo Rech

https://doi.org/10.33434/cams.1064713

Öz

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.

Anahtar Kelimeler

Chaos, Lyapunov exponents spectrum, Parameter space, Period-adding bifurcation

Destekleyen Kurum

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

Proje Numarası

0

Kaynakça

[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.

Yıl 2022, , 92 - 98, 30.06.2022

Angela Da Silva Paulo Rech

https://doi.org/10.33434/cams.1064713

Öz

Proje Numarası

0

Kaynakça

[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.

Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Makaleler
Yazarlar	Angela Da Silva Paulo Rech
Proje Numarası	0
Yayımlanma Tarihi	30 Haziran 2022
Gönderilme Tarihi	31 Ocak 2022
Kabul Tarihi	27 Haziran 2022
Yayımlandığı Sayı	Yıl 2022

Kaynak Göster

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. Haziran 2022;5(2):92-98. doi:10.33434/cams.1064713
Chicago	Da Silva, Angela, ve Paulo Rech. “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”. Communications in Advanced Mathematical Sciences 5, sy. 2 (Haziran 2022): 92-98. https://doi.org/10.33434/cams.1064713.
EndNote	Da Silva A, Rech P (01 Haziran 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 ve P. Rech, “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”, Communications in Advanced Mathematical Sciences, c. 5, sy. 2, ss. 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 (Haziran 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 ve Paulo Rech. “On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network”. Communications in Advanced Mathematical Sciences, c. 5, sy. 2, 2022, ss. 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.