Patterns in hierarchical networks of neuronal oscillators with \(\mathbb D_3\times \mathbb Z_3\) symmetry
The authors study a hierarchical delay-coupled network with a \(\mathbb{D}_3\times\mathbb{Z}_3\)-symmetry. Its dynamics is given by \[ \dot{x}_{(i,j)}(t) = - x_{(i,j)}(t) + af(x_{(i,j-1)}(t-\tau)) + bg(x_{(i+1,j)}(t-\tau)) + bg(x_{(i-1,j)}(t-\tau)), \] where all indices are considered modulo 3 and the sigmoid functions \(f\) and \(g\) are smooth. This system can be considered as a bidirectional ring consisting of three unidirectional rings. Explicit conditions for linear stability of the trivial solution and for its bifurcations are given. The codimension-one transcritical and Hopf bifurcations and the symmetries of the emerging states are classified. Further, codimension-two bifurcations are studied via center manifold reductions and possible generalizations are discussed.
scientific article; zbMATH DE number 6154869