Parametric Resonance in a dissipative system \'a la Kronig-Penney

The competition between parametric resonance (PR) and dissipation is studied in the damped Kronig-Penney model, with time-dependent dissipation rate gamma(t). In the classical case, it is shown that dissipation leaves just a finite number of PR-bands at most, suppressing those at higher frequencies. An analysis of the Lewis-Reisenfeld invariant I(q,p,rho) is performed, showing that, in the PR regime, the auxiliary function rho(t) can be chosen bounded or unbounded, depending on the initial conditions.