Instability of rapidly-oscillating periodic solutions for discontinuous differential delay equations

Abstract

We study the equation $$ (\star)\qquad \dot{x}(t)=- h(x(t-1)) + f(x(t)) \ \text{ for } \ t\ge 0, \ x_{|_{[-1,0]}}=x_0, $$ where $h$ is an odd function defined by $h(y)$ is equal to $a$ if $0 < y <c$, equal to $b$ if $y \ge c$, $a>b>0$ and $c>0$ and $f$ is an odd ${{\mathcal C}}^1$ function such that $\sup |f(x)| <b$. We first consider the equation $\dot{x}(t)=- h(x(t-1))$, corresponding to $f\equiv 0$. We find the admissible shapes of rapidly-oscillating symmetric periodic solutions and we show that these periodic solutions are all unstable. We then extend these results to our general equation $(\star)$.