TY - JOUR
AU - Couchouron, Jean-Francois
AU - Kamenskii, Mikhail I.
AU - Mikhaylenko, Boris
AU - Nistri, Paolo
PY - 2015/12/01
Y2 - 2023/04/01
TI - Periodic bifurcation problems for fully nonlinear neutral functional differential equations via an integral operator approach: the multidimensional degeneration case
JF - Topological Methods in Nonlinear Analysis
JA - TMNA
VL - 46
IS - 2
SE -
DO - 10.12775/TMNA.2015.062
UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2015.062
SP - 631 - 664
AB - 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.
ER -