Search

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 \hbox{$P(x,\varepsilon) +\varepsilon Q(x, \varepsilon)=0$.} 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.

Resume Nous prouvons l'unicite des solutions pour une classe d'equations differentielles ordinaires gouvernees par des deriveurs (i.e. l'oppose est quasi monotone relativement a ( R + ) N ) continus sur R N et pour laquelle les conditions d'accretivite ou les theoremes de Nagumo–Osgood–Kamke ne peuvent s'appliquer.

In this paper we consider an infinite dimensional bifurcation equation depending on a parameter $ \varepsilon>0 . $ By means of the theory of condensing operators, we prove the existence of a branch of solutions, parametrized by $ \varepsilon , $ bifurcating from a curve of solutions of the bifurcation equation obtained for $\varepsilon =0 . $ We apply this result to a specific problem, namely to the existence of periodic solutions bifurcating from the limit cycle of an autonomous functional differential equation of neutral type when it is periodically perturbed by a nonlinear perturbation term of small amplitude.