Metastable periodic patterns in singularly perturbed state dependent delayed equations

We consider the scalar delayed differential equation $\ep\dot x(t)=-x(t)+f(x(t-r))$, where $\ep>0$, $r=r(x,\ep)$ and $f$ represents either a positive feedback $df/dx>0$ or a negative feedback $df/dx<0$. When the delay is a constant, i.e. $r(x,\ep)=1$, this equation admits metastable rapidly oscillating solutions that are transients whose duration is of order $\exp(c/\ep)$, for some $c>0$. In this paper we investigate whether this metastable behavior persists when the delay $r(x,\ep)$ depends non trivially on the state variable $x$. Our conclusion is that for negative feedback, the persistence of the metastable behavior depends only on the way $r(x,\ep)$ depends on $\ep$ and not on the feedback $f$. In contrast, for positive feedback, for metastable solutions to exist it is further required that the feedback $f$ is an odd function and the delay $r(x,\ep)$ is an even function. Our analysis hinges upon the introduction of state dependent transition layer equations that describe the profiles of the transient oscillations. One novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant.