Repressilator

A repressilator is a cyclic negative-feedback gene regulatory network in which three repressors inhibit one another in sequence. In a normalized logic approximation for gene regulation, $x^{\prime }=H(h-z)-x$, $y^{\prime }=H(h-x)-y$, $z^{\prime }=H(h-y)-z$. For $0<h<1$, threshold crossings successively switch production and can generate sustained oscillations. A rise from $v_0<h$ reaches threshold after $\ln [(1-v_0)/(1-h)]$; a decay from $v_0>h$ reaches it after $\ln (v_0/h)$ in units of the common decay time.