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Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 931–946 | Cite as

Neimark–Sacker Bifurcation with \(\mathbb {Z}_n\)-Symmetry and a Neural Application

  • Reza Mazrooei-SebdaniEmail author
  • Zohreh Eskandari
Article
  • 24 Downloads

Abstract

Effects of \(\mathbb {Z}_n\)-symmetry, (\(n\ge 2\)), on normal form of Neimark–Sacker bifurcation in discrete time dynamical systems are investigated. As an application, we consider three dimensional discrete Hopfield neural network with \(\mathbb {Z}_2\)-symmetry. We drive analytical conditions for stability and bifurcations of the trivial fixed point of the system and compute analytically the normal form coefficients for the codimension 1 and codimension 2 bifurcation points including pitchfork, period-doubling, Neimark–Sacker, \(\mathbb {Z}_2\)-symmetric Neimark–Sacker and resonance 1:4. By using numerical continuation in numerical software matcontm, we compute bifurcation curves of trivial fixed point and cycle with period 4 under variation of one and two parameters, and all codimension 1 and codimension 2 bifurcations supported by matcontm, on the corresponding curves are computed.

Keywords

Neimark–Sacker bifurcation \(\mathbb {Z}_n\)-symmetry Map Critical normal form coefficient Numerical continuation Hopfield neural network 

Notes

References

  1. 1.
    Chossat, P., Golubitsky, M.: Iterates of maps with symmetry. SIAM J. Math. Anal. 19, 1259–1270 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A., Meijer, H.G.E., Sautois, B.: New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14, 147–175 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dong, J.-Y., Zhang, J.-Y.: Several symmetry properties of discrete Hopfield neural networks. In: Proceedings of International Conference on Machine Learning and Cybernetics, Beijing 4–5 Nov 2002, pp. 1374–1378Google Scholar
  4. 4.
    Field, M.J.: Dynamics and Symmetry. Advanced Texts in Mathematics, vol. 3. Imperial College Press, London (2007)CrossRefGoogle Scholar
  5. 5.
    Goles, E., Matamala, M.: Symmetric discrete universal neural networks. Theor. Comput. Sci. 168, 405–416 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, Vol. I. Applied Mathematical Sciences, vol. 51. Springer, New York (1985)CrossRefGoogle Scholar
  7. 7.
    Golubitkky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, Vol. II. Applied Mathematical Sciences, vol. 69. Springer, New York (1988)CrossRefGoogle Scholar
  8. 8.
    Govaerts, W., Khoshsiar Ghaziani, R., Kuznetsov, Y.A., Meijer, H.G.E.: Numerical methods for two parameter local bifurcation analysis of maps. SIAM. J. Sci. Comput. 29, 2644–2667 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Iooss, G., Adelmeyer, M.: Topics in Bifurcation Theory and Applications. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  10. 10.
    Kaslik, E., Balint, S.: Bifurcation analysis of a two-dimentional delayed discrete-time hopfield neural network. Chaos Soliton Fractals 34, 1245–1253 (2007)CrossRefGoogle Scholar
  11. 11.
    Kaslik, E., Balint, S.: Complex and chaotic dynamics in a discrete-time-delayed hopfield neural network with ring architecture. Neural Netw. 22, 1411–1418 (2009)CrossRefGoogle Scholar
  12. 12.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, Berlin (2004)CrossRefGoogle Scholar
  13. 13.
    Kuznetsov, Y.A., Meijer, H.G.E.: Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues. SIAM. J. Sci. Comput. 26, 1932–1954 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mazrooe-Sebdani. R.: On bifurcations of delay single species discrete time population models. J. Differ. Equ. Appl. 24, 192–219 (2018)Google Scholar
  15. 15.
    Mazrooei-Sebdani, R., Eskandari, Z., Meijer, H.G.E.: Numerical bifurcation analysis of double \(+1\) multiplier in \(\mathbb{Z}_3\)-symmetric maps, TW Memorandum 2058, Department of Mathematics, University of Twente. http://www.math.utwente.nl/publications/. Accessed 7 Mar 2017
  16. 16.
    Mazrooei-Sebdani, R., Eskandari, Z., Meijer, H.G.E.: Numerical bifurcation analysis of double \(+1\) multiplier in \(\mathbb{Z}_3\)-symmetric maps. J. Differ. Equ. Appl. 24, 1–12 (2018)CrossRefGoogle Scholar
  17. 17.
    Mazrooei-Sebdani, R., Eskandari, Z.: Numerical Analysis of Normal Forms for Double Multipliers Bifurcations with a Refection Symmetry, revised (2019)Google Scholar
  18. 18.
    Meijer, H.G.E.: Codimension 2 Bifurcations of Iterated Maps. Ph.D. Thesis, Utrecht University (2006)Google Scholar
  19. 19.
    Peng, M., Yuan, Y.: Synchronization and desynchronization in a delayed discrete neural network. Int. J. Bifurc. Chaos 17, 791–803 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shu, H., Wei, J.: Bifurcation analysis in a discrete BAM network model with delays. J. Differ. Equ. Appl. 17, 69–84 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, Ch., Zheng, B.: Three cell symmetry discrete-time-delayed neural network. Neurocomputing 122, 239–244 (2013)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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