Abstract
In the present work, we investigate phase correlations by recourse to the Shannon entropy. Using theoretical arguments, we show that the entropy provides an accurate measure of phase correlations in any dynamical system, in particular when dealing with a chaotic diffusion process. We apply this approach to different low-dimensional maps in order to show that indeed the entropy is very sensitive to the presence of correlations among the successive values of angular variables, even when it is weak. Later on, we apply this approach to unveil strong correlations in the time evolution of the phases involved in the Arnold’s Hamiltonian that lead to anomalous diffusion, particularly when the perturbation parameters are comparatively large. The obtained results allow us to discuss the validity of several approximations and assumptions usually introduced to derive a local diffusion coefficient in multidimensional near-integrable Hamiltonian systems, in particular the so-called reduced stochasticity approximation.
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Notes
By weak chaos we mean the dynamical state when the unstable chaotic motion is mostly confined to the narrow stochastic layers around resonances.
In fact, the dynamical system could involve more than one action and phase.
See the discussion below.
This could occur due to the smallness of the time step.
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Acknowledgements
This work was supported by Grants from Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET), the Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata. We acknowledge two anonymous reviewers for their valuable comments and suggestions that allow us to improve this manuscript.
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Cincotta, P.M., Giordano, C.M. Phase correlations in chaotic dynamics: a Shannon entropy measure. Celest Mech Dyn Astr 130, 74 (2018). https://doi.org/10.1007/s10569-018-9871-3
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DOI: https://doi.org/10.1007/s10569-018-9871-3