Combining machine learning with recurrence analysis for resonance detection

The width of a resonance in a nearly integrable system, i.e., in a nonintegrable system where chaotic motion is still not prominent, can tell us how a perturbation parameter is driving the system away from integrability. Although the tool that we are presenting here can be used, is quite generic, and can be used in a variety of systems, our particular interest lies in binary compact object systems known as extreme mass ratio inspirals (EMRIs). In an EMRI, a lighter compact object, like a black hole or a neutron star, inspirals into a supermassive black hole due to gravitational radiation reaction. During this inspiral, the lighter object crosses resonances, which are still not very well modeled. Measuring the width of resonances in EMRI models allows us to estimate the importance of each perturbation parameter able to drive the system away from resonances and decide whether its impact should be included in EMRI waveform modeling or not. To tackle this issue in our study, we show first that recurrence quantifiers of orbits carry imprints of resonant behavior, regardless of the system's dimensionality. As a next step, we apply a long short-term memory machine learning architecture to automate the resonance detection procedure. Our analysis is developed on a simple standard map and gradually we extend it to more complicated systems, until finally we employ it in a generic deformed Kerr spacetime known in the literature as the Johannsen-Psaltis spacetime.