**Author details**

**7. Conclusions**

*Progress in Relativity*

We have presented a detailed analysis of the stability indicators LE, RE, and REM recently proposed. Defining the square of LE as the trace of the tangent map times, its transpose renders this indicator independent from the choice of an initial vector, which can introduce spurious structures. The RE is the reversibility error due to additive random noise, whereas REM is the reversibility error due to the roundoff. A very simple relation is found between RE and LE. The oscillations, which affect the fast Lyapunov indicator, can be filtered with MEGNO. Since RE has a smooth behavior and does not exhibit oscillations, filtering it by MEGNO is not necessary. The asymptotic behavior of REM is similar to RE even though it exhibits large fluctuations. The displacements caused by roundoff are almost random vectors, if the map has a high computational complexity, but since just a single

realization of the process is available, the fluctuations cannot be averaged. We have first examined the behavior of LE and RE for linear maps and for integrable maps. If the fixed point is elliptic, then the asymptotic growth follows a power law *n<sup>α</sup>*, and the exponent does not depend on the chosen coordinates for LE and RE. Conversely, the presence of oscillations and their amplitude depends on the choice of coordinates. The growth of REM also follows a power law, but the choice

by symplectic maps, with algorithms which provide simultaneously the

corresponding tangent maps [20, 21], in order to compute the errors discussed so far. A special care is required in comparing RE with REM when the chosen phase plane is invariant. Indeed given an initial point in the invariant plane, the noise brings the orbit out of it, whereas the roundoff usually does not. In this case a random kick before reversing the orbit is sufficient to bring the orbit out of the invariant plane and to restore the correspondence between REM and RE. The satisfactory results obtained so far, not only in the simple models presented here but also in high dimensional models of celestial mechanics, prove that the method we

For a generic map which has regular and chaotic components, the error growth follows a power law and an exponential law, respectively. For the standard map and the Hénon map, the behavior of LE, RE, and REM has been compared first by varying the iteration order *n* up to a some value *N*, for a selected initial condition. Then the errors for *n* ¼ *N* have been compared when the initial point moves on a line. The theoretical predictions concerning the power law growth in the regular regions and the exponential growth in the chaotic ones are confirmed. For twodimensional maps, the error plots for initial conditions in a rectangular domain of phase space are very similar, and the correspondence with the phase space portraits is excellent. Moreover, the different plots allow a quantitative comparison of the orbital sensitivity to initial displacements, noise, and roundoff. For maps of dimension 4 or higher, the proposed error plots on selected phase planes allow to inspect the orbital stability. Hamiltonian flows must be approximated with a high accuracy

of coordinates affects the exponent itself.

propose has a wide range of applicability.

**178**

Giorgio Turchetti<sup>1</sup> \* and Federico Panichi<sup>2</sup>

1 Department of Physics and Astronomy, Alma Mater Studiorum—University of Bologna, Bologna, Italy

2 Faculty of Mathematics and Physics, Institute of Physics, University of Szczecin, Szczecin, Poland

\*Address all correspondence to: giorgio.turchetti@unibo.it

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
