Hysteresis and Differential Equations

Many dynamical systems exhibit hysteresis as one of their features. In classical continuum mechanics, hysteretic behaviour is inherent in many constitutive laws. In systems and control applications, hysteresis regularly appears via mechanical play and friction, or in the form of a relay or thermostat, often deliberately built into the system. If the hysteretic behaviour is described using a hysteresis operator, then the mathematical model for the dynamical system consists of a system of differential equations coupled with one or several hysteresis operators, which is complemented by initial and boundary conditions. The oscillator with hysteretic restoring force,$$ y''(t) + W[y](t) = f(t), $$W being a hysteresis operator, furnishes a basic example1 2.