  Workhorse theories throughout all of physics derive effective Hamiltonians to
describe slow time evolution, even though low-frequency modes are actually
coupled to high-frequency modes. Such effective Hamiltonians are accurate
because of \textit{adiabatic decoupling}: the high-frequency modes `dress' the
low-frequency modes, and renormalize their Hamiltonian, but they do not
steadily inject energy into the low-frequency sector. Here, however, we
identify a broad class of dynamical systems in which adiabatic decoupling fails
to hold, and steady energy transfer across a large gap in natural frequency
(`steady downconversion') instead becomes possible, through nonlinear
resonances of a certain form. Instead of adiabatic decoupling, the special
features of multiple time scale dynamics lead in these cases to efficiency
constraints that somewhat resemble thermodynamics.
