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

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.