https://doi.org/10.1140/epjs/s11734-023-01017-x
Regular Article
Chaotic diffusion in the action and frequency domains: estimate of instability times
1
Instituto de Astronomia, Geofísica e Ciências Atmosférias, Universidade de São Paulo, Rua do Matão, 1226, Butantã, 05508-090, São Paulo, São Paulo, Brazil
2
Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033, Tokyo, Japan
a gabriel.guimaraes@usp.br, gabriel.guimaraes@grad.nao.ac.jp
Received:
15
May
2023
Accepted:
9
November
2023
Published online:
7
December
2023
Chaotic diffusion in non-linear systems is commonly studied in the action framework. In this paper, we show that its study in the frequency domain provides good estimates of the sizes of the chaotic regions in the phase space, as well as the diffusion timescales inside these regions. Applying traditional tools, such as Poincaré Surfaces of Section, Lyapunov Exponents and Spectral Analysis, we characterise the phase space of the Planar Circular Restricted Three Body Problem (PCR3BP). For the purpose of comparison, the diffusion coefficients are obtained in the action domain of the problem, applying the Shannon Entropy Method (SEM), as well as in the frequency domain, applying the Mean Squared Displacement (MSD) method and Laskar’s Equation of Diffusion. We compare the diffusion timescales defined by the diffusion coefficients obtained to the Lyapunov times and the instability times obtained through direct numerical integrations. Traditional tools for detecting chaos tend to misrepresent regimes of motion, in which either slow-diffusion or confined-diffusion processes dominates. The SEM shows a good performance in the regions of slow chaotic diffusion, but it fails to characterise regions of strong chaotic motion. The frequency-based methods are able to precisely characterise the whole phase space and the diffusion times obtained in the frequency domain present satisfactory agreement with direct integration instability times, both in weak and strong chaotic motion regimes. The diffusion times obtained by means of the SEM fail to match correctly the instability times provided by numerical integrations. We conclude that the study of dynamical instabilities in the frequency domain provides reliable estimates of the diffusion timescales, and also presents a good cost-benefit in terms of computation-time.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
