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


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.


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



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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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