Local hysteresis in relaxation oscillators

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Abstract
Relaxation oscillations are very commonly found in nature. When modulation by an external force is present, a global hysteresis connected with chaos is often encountered. In addition to this kind of hysteresis a qualitatively different ‘‘local’’ form is found in some systems. We describe the circumstances under which one observes 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 the driven damped-pendulum equation, the hysteresis is limited in size as compared to the phase-locked region.