This paper considers a class of hysteresis systems consisting of a linear part with an external input and feedback with a backlash nonlinearity. Assuming that the latter is specified by a strongly convex set, we establish estimates for the Lyapunov exponents which quantify the rate of convergence of the system state trajectories to a forced periodic regime when the input is a periodic function of time with a sufficiently large "amplitude". These results employ enhanced dissipation inequalities, arising from differential inclusions with strongly convex sets which were used previously for the Moreau sweeping process.