Dynamical stability of quasi-periodic response solutions in planar conservative systems

We study non-autonomous planar Hamiltonian or reversible vector fields that vanish at the origin. The time-dependence is quasi-periodic with strongly non-resonant frequencies. First, we give a simple criterion in terms of the averaged system for the trivial solution to be dynamically stable. Then we obtain destabilizations for classes of examples where the conditions of the criterion are not satisfied. We end with possible ways to stabilize an unstable trivial solution by means of vector fields with zero average.