Periodic bifurcation problems for fully nonlinear neutral functional differential equations via an integral operator approach: the multidimensional degeneration case

Abstract

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 $P(x,\varepsilon) +\varepsilon Q(x, \varepsilon)=0$. 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 $P'(x_0(\theta),0)=0$, $\theta\in[0,T]$. 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.