  We study resonances of multidimensional chaotic map dynamics. We use the
calculus of variations to determine the additive forcing function that induces
the largest response, that is, the greatest deviation from the unperturbed
dynamics. We include the additional constraint that only select degrees of
freedom be forced, corresponding to a very general class of problems in which
not all of the degrees of freedom in an experimental system are accessible to
forcing. We find that certain Lagrange multipliers take on a fundamental
physical role as the efficiency of the forcing function and the effective
forcing experienced by the degrees of freedom which are not forced directly.
Furthermore, we find that the product of the displacement of nearby
trajectories and the effective total forcing function is a conserved quantity.
We demonstrate the efficacy of this methodology with several examples.
