Local hysteresis in relaxation oscillators

Relaxation oscillations or 'integrate and fire' phenomena are very commonly found in nature. When modulated by an external force a global hysteresis connected with chaos is often encountered. Besides this kind of hysteresis a local form is found in some systems. We describe briefly the difference and the circumstances under which to observe local hysteresis. A specific system treated in detail is the Fohlmeister model, originally derived to describe a neuronal encoder. In the limit of small damping an analytical solution is obtained. Furthermore, we derive an upper limit to the hysteresis. The results are compared to numerical calculations on the full system and agree quite well. In contrast to e.g. the driven damped pendulum equation the hysteresis is limited in size as compared to the phase-locked region. (orig.)