Graceful switching in hybrid models

Classical oscillators, such as the ideal pendulum and the harmonic oscillator, can be modeled as switching systems. In these models, the switching takes place in a ¿graceful¿ fashion when the rate of displacement is zero. Such examples raise the question of whether hybrid models with such ¿graceful¿ switching can be modeled using ordinary differential equations, and of how to identify cases of hybrid systems which admit alternative classical descriptions in higher dimensions. Conversely, in a similar spirit, we discuss how a relaxation oscillator which consists of an integrator in feedback with an ideal relay-hysteresis can be approximated by a second order system.