Classification of the Dynamical Behaviour of a Piecewise Oscillator

We consider a forced damped oscillator whose motion is modeled by a second-order non-autonomous piecewise linear differential equation. The system has a continuous regime, where the time flow is characterized by the explicit solutions of ordinary differential equations, and a singular regime, where the time flow is characterized by an appropriate transformation relating the explicit solutions from one domain to another. Therefore it can be classified as an hybrid system. We determine the regions of the parameter space where the Poincaré map, which describes the observed motion of the forced damped oscillator, can be classified as m−modal map, for given m natural.