Persistence and periodic solutions in systems of delay differential equations

PABLO AMSTER; MELANIE BONDOREVSKY

Amsterdam

Congreso; Dynamics Days Digital 2020; 2020

Vrije Universiteit Amsterdam

We study semi-dynamical systems associated to delay differential equations:\begin{equation}\label{eq}x'(t)=f(t,x(t),x(t-\tau))\end{equation}where$f:\R\times [0,+\infty)^{2N}\to \R^N$ An initial condition for can be expressed in the following way\begin{equation}\label{icsd}x_0=\varphi,\end{equation}where $\varphi:[-\tau,0]\to [0,+\infty)^N$ is a continuous function and $x_t\in C([-\tau,0],\R^N)$ is defined by $x_t(s)=x(t+s)$. Thus, the flow \begin{equation} \label{flow} \Phi:[0,+\infty)\times C([-\tau,0],\R^N)\to C([-\tau,0],\R^N),\end{equation} given by $\Phi(t,\varphi)=x_t$, induces a semi-dynamical system.We give sufficient conditions to guarantee uniform persistence, employing guiding functions techniques.In order to find periodic orbits of (\ref{flow}) we employ topological degree methods. Since the space of initial conditions is infinite dimensional, the Brouwer degree cannot be applied: we instead use Leray-Schauder degree techniques.