Deterministic and Stochastic Surrogate Models for a Slowly Driven Fast Oscillator

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1090-1107, June 2024.
Abstract.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 that, in the context of a fast harmonic oscillator forced by one component of the Lorenz 1963 model, this principle can be used to construct time-discrete surrogate models by numerically extracting approximate locations and residues of complex poles via adaptive Antoulas–Anderson (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 analogue of Gottwald and Melbourne’s 2004 “0–1 test for chaos”; that is, the asymptotic growth rate of the energy in the oscillator can discern whether or not the forcing function is chaotic.