Analysis of Round Off Errors With Reversibility Test as a Dynamical Indicator

Abstract

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analysed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator.The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same.Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.

Extremes and Recurrence in Dynamical Systems

“…We first introduce two standard methods used for this purpose, the analysis of the divergence of orbits due to the numerical round-off and the reversibility error. We briefly summarize here some definitions and suggest the reader to look into [244] for further clarifications.…”

“…This is in agreement with the idea that we should get EVL for infinitely small noises in the limit of infinitely long samples. In our case, EVLs are obtained only for > 10 −4 , which is still considerably larger than the noise introduced by round-off resulting from double precision, as the round-off procedure is equivalent to the addition to the exact map of a random noise of order 10 −7 [243,244]. This suggests that is relatively hard to get rid of the properties of the underlying deterministic dynamics just by adding some noise of unspecified strength and considering generically long time series: the emergence of the smoothing due to the stochastic perturbations is indeed non-trivial when considering very local properties of the invariant measure as we do here.…”

“…The quality of the fitting improves when larger bins are considered: this is in agreement with the idea that we should get EVL for infinitely small noises in the limit of infinitely long samples. In our case, EVLs are obtained only for ε > 10 −4 , which is still considerably larger than the noise introduced by round-off resulting from double precision, as the round-off procedure is equivalent to the addition to the exact map of a random noise of order 10 −7 [243,244]. …”

A lower bound error for free-run simulation of the polynomial NARMAX

“…In that work, a careful way is presented to calculate error propagation in the computation of recursive function. Investigation of propagation error is not a recent issue (Galias, 2013;Faranda, Mestre, & Turchetti, 2012;Goldberg, 1991;Hammel et al, 1987). In fact, there are many works based on deterministic or stochastic tools that provide some confidence in simulation of recursive functions.…”

“…To deal with the numerical questions in nonlinear dynamic systems, many researchers have used the shadowing property (Chow & Palmer, 1992, 1991Hammel et al, 1987;Sauer, Grebogi, & Yorke, 1997). This property ensures that a pseudo-orbit (a sequence of points calculated from a recursive function map) is a homeomorphism that remains close to the real orbit (Faranda et al, 2012). What fewer researchers report is the fact that this property is valid only for uniformly hyperbolic systems (Lozi, 2013).…”