  It has long been known that the excitation of fast motion in certain
two-scale dynamical systems is linked to the singularity structure in complex
time of the slow variables. We demonstrate, in the context of a fast harmonic
oscillator forced by one component of the Lorenz 1963 model, that this
principle can be used to construct time-discrete surrogate models by
numerically extracting approximate locations and residues of complex poles via
Adaptive Antoulas-Andersen (AAA) rational interpolation and feeding this
information into the known "connection formula" to compute the resulting fast
amplitude. Despite small but nonnegligible local errors, the surrogate model
maintains excellent accuracy over very long times. In addition, we observe that
the long-time behavior of fast energy offers a continuous-time analog of
Gottwald and Melbourne's 2004 "0-1 test for chaos" - the asymptotic growth rate
of the energy in the oscillator can discern whether or not the forcing function
is chaotic.
