arXiv : Performance analysis of indicators of chaos for nonlinear dynamical systems

Bazzani, A.; Turchetti, G.; Montanari, C.E.; Giovannozzi, M.

doi:10.1103/PhysRevE.107.064209

Article
Report number	arXiv:2304.08340
Title	Performance analysis of indicators of chaos for nonlinear dynamical systems
Author(s)	Bazzani, A. (Bologna U. ; INFN, Bologna) ; Giovannozzi, M. (CERN) ; Montanari, C.E. (Bologna U. ; INFN, Bologna ; CERN) ; Turchetti, G. (Bologna U. ; INFN, Bologna)
Publication	2023-06-22
Imprint	2023-04-13
Number of pages	24
In:	Phys. Rev. E 107 (2023) 064209
DOI	10.1103/PhysRevE.107.064209 (publication)
Subject category	physics.acc-ph ; Accelerators and Storage Rings ; nlin.CD ; Nonlinear Systems ; math.DS ; Mathematical Physics and Mathematics
Abstract	The efficient detection of chaotic behavior in orbits of a complex dynamical system is an active domain of research. Several indicators have been proposed in the past, and new ones have recently been developed in view of improving the performance of chaos detection by means of numerical simulations. The challenge is to predict chaotic behavior based on the analysis of orbits of limited length. In this paper, the performance analysis of past and recent indicators of chaos, in terms of predictive power, is carried out in detail using the dynamical system characterized by a symplectic Hénon-like cubic polynomial map.
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0) publication: Published by the American Physical Society (License: CC-BY-4.0)

$Survival plot for various $4d$ modulated Hénon maps with quadratic and cubic nonlinearities. Initial conditions, sampled on an uniform Cartesian $300\times300$ grid in the $x-y$ plane, are tracked up to $n_\text{max}=10^8$ and are considered lost when their distance to the origin exceeds a predefined maximum radius $r_\mathrm{c}=10^2$. The two sets of linear frequencies feature different shapes of the stable region as can be seen by comparing the plots in the two rows. The parameters $\varepsilon$ and $\mu$ induce additional changes, in particular the increase of the size of the transition region between stability up to $n_\mathrm{max}$ and shorter stability time. The color scale is related to the logarithm of the stability time as reported on the right.$ $Color maps of $\log_{10}(\log_{10}(LE)/n)$ and $\log_{10}(FLI/n)$ indicators for a low iteration number ($n=10^2$, top row) and a high iteration number ($n=10^4$, bottom row). In both rows, the $FLI$ for the four possible displacements is shown together with $LE$, to highlight the different structures shown by the indicators. It is possible to see how, for low iteration numbers, different choices of initial displacement for $FLI$ highlight structures that do not appear in $LE$. The differences reduce for higher number of turns, but are still present. These structures have a vage resemblance with the net of resonances that is present in phase space. Note that an arrow at the bottom of the color bar means that pixels of the bottom color correspond to a value equal to or lower than the bottom value. White pixels correspond to initial conditions whose distance from the origin has exceeded a predefined radius ($r_c=10^2$) during the tracking, before reaching the target iteration number $n$. (Simulation parameters used: $(\omega_{x0},\omega_{y0})= (0.168,\ 0.201),\ \varepsilon=64.0,\ \mu=0.5$).$ $Time evolution of $FLI$ computed using either a standard mean or the Birkhoff averaging. Top plot: indicators computed for a set of 100 regular initial conditions, the fit highlights a faster convergence rate for the Birkhoff averaging. Bottom plot: indicators computed for a set of 100 chaotic initial conditions. A similar improvement in convergence rate is observed for low $n$ values, before reaching a saturation value of the indicator of the order of $10^{-3}$. (Simulation parameters: $(\omega_{x0},\omega_{y0})= (0.28,\ 0.31),\ \varepsilon=32.0,\ \mu=0.5$).$ Show more plots

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