Combining Machine Learning with Recurrence Analysis for resonance detection

TL;DR

The paper addresses the challenge of detecting orbital resonances in weakly nonintegrable dynamical systems, with a specific aim of informing EMRI waveform modeling. It introduces a hybrid approach that combines recurrence quantification analysis with a bidirectional LSTM to automate resonance detection across low- and high-dimensional systems. The method is validated on a Standard map-based training regime and tested on the de Vogeleare map, Poincaré sections around a perturbed Kerr black hole, and a 4D map, showing robust resonance localization in several cases and highlighting limitations in higher dimensions. The work offers a practical framework for resonance-aware modeling in relativistic astrophysical contexts and suggests directions for improving embedding strategies and training data to handle complex phase spaces.

Abstract

The width of a resonance in a nearly integrable system, i.e. in a non-integrable 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.

Figures (10)

• Figure 1: Phase portrait of the standard map (see Eqs. \ref{['eqs:smap']}) with $K=0.8$, showing the standard hallmarks of chaotic maps: KAM curves, Birkhoff chains, and chaotic layers. The regular, resonant, and chaotic orbits which produce the rp of Fig. \ref{['fig:example_rps']} are highlighted in red, blue, and green, respectively.
• Figure 2: Examples of recurrence plots of trajectories of the standard map with $K=0.8$. Different recurrence thresholds are used in order to achieve a fixed $\mathrm{RR} = 0.05$, and all three are generated using the Euclidean metric. a) KAM trajectory starting at $\left(x_0,\, y_0\right) = \left(\pi,\, -4\pi/5\right)$, using $\epsilon = 0.16471$. b) Resonant trajectory starting at $\left(x_0,\, y_0\right) = \left(\pi,\, -37\pi/50\right)$, using $\epsilon = 0.045881$. c) Chaotic trajectory starting at $\left(x_0,\, y_0\right) = \left(\pi,\, -27\pi/50\right)$, using $\epsilon = 0.286985$.
• Figure 3: Example of curves of the standard map at $K=0.8$ as a function of the initial condition taken along the $y$-axis of Fig. \ref{['fig:example_smap']} with $x_0=0$. Each of the three panel corresponds to one of the recurrence quantifiers: the recurrence rate, the laminarity, and the entropy of the diagonal lines distribution, in Eqs. \ref{['eq:def_rr']}, \ref{['eq:def_lam']}, \ref{['eq:def_l_entr']}, respectively. The color-coding corresponds to different recurrence thresholds given in the legend. The pale blue areas in the background show the locations of resonances as marked during the training data labeling procedure.
• Figure 4: network architecture. Taken from Alfaidi:2024ioo.
• Figure 5: Geography of the resonances of the 4D map in the $x$, $z$ plane. The arrows on the right side show the $z$ values corresponding to the initial conditions taken in Figs. \ref{['fig:test_4d_map']} and \ref{['fig:test_4d_map_embed']}. The choice of the color mapping between 0.4 and 1 precludes us from identifying regular and chaotic trajectories individually; however, it clearly shows thin chaotic layers at the edges of resonances. The APLE is computed on a $501\times 501$ grid and the perturbation parameter is set to $K=0.05$.