Multiple slowly oscillating periodic solutions in coupled lossless transmission lines
Using equivariant degree theory [cf. \textit{W. Krawcewicz} and \textit{J. Wu}, Theory of degrees with applications to bifurcations and differential equations (1997; Zbl 0882.58001)] the article studies symmetric periodic solutions in neutral functional differential equations with dihedral symmetry depending on one parameter. The authors first obtain symmetric Hopf bifurcation theorems in this general setting and then apply these theorems to a polygonal ring of diffusively coupled identical oscillators. The article studies existence, multiplicity and global continuations of branches of periodic solutions. In particular, it proves the existence of large amplitude synchronous (all cells have always equal value) and phase-locked (all cells oscillate identically but not with the same phase) periodic solutions in these ring-structured neutral systems. A concrete example illustrates the coexistence of a large number of slowly oscillating solutions. The problem of stability of these periodic solutions is, naturally, left open.
neutral functional differential equation
scientific article; zbMATH DE number 2076993