Equivariant Hopf-Pitchfork Bifurcation of Symmetric Coupled Neural Network with Delay

This paper is concerned with how the symmetry and singularity of a system of differential equations affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem satisfying a D3-symmetry condition, the mode interactions in two-parameter bifurcations with a single zero and two pairs of imaginary roots are considered. The codimension two normal form with equivariant Hopf-pitchfork bifurcations are given. Through analyzing the unfolding structure, local classification in the neighborhood of equivariant Hopf-pitchfork bifurcations point for the D3-symmetric is undertaken. A rich variety of dynamical and bifurcation behaviors is pointed out. Beyond a stable fixed point or a pair of stable fixed points, some interesting phenomena are also found, such as the coexistence of two periodic solutions which are verified both theoretically and numerically.

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