Asymptotic stability equals exponential stability, and ISS equals finite energy gain---if you twist your eyes

In this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables. The same is shown for input-to-state stability and input-to-state exponential stability, and for input-to-state exponential stability and a nonlinear $H_{\mathrm{\infty }}$ estimate.