Periodic bifurcation problems for fully nonlinear neutral functional differential equations via an integral operator approach: the multidimensional degeneration case - Couchouron, Jean-Francois; Kamenskii, Mikhail I.; Mikhaylenko, Boris; Nistri, Paolo (Topological Methods in Nonlinear Analysis, 2015)
We consider a T-periodically perturbed autonomous functional differential equation of neutral type. We assume the existence of a T-periodic limit cycle x_0 for the unperturbed autonomous system. We also assume that the linearized unperturbed equation around the limit cycle has the characteristic multiplier 1 of geometric multiplicity 1 and algebraic multiplicity greater than 1. The paper deals with the existence of a branch of T-periodic solutions emanating from the limit cycle. The problem of finding such a branch is converted into the problem of finding a branch of zeros of a suitably defined bifurcation equation P(x,ε) +ε Q(x, ε)=0. The main task of the paper is to define a novel equivalent integral operator having the property that the T-periodic adjoint Floquet solutions of the unperturbed linearized operator correspond to those of the equation P’(x_0(θ),0)=0, θ[0,T]. Once this is done it is possible to express the condition for the existence of a branch of zeros for the bifurcation equation in terms of a multidimensional Malkin bifurcation function.