Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1910.11976

On the risks of using double precision in numerical simulations of spatio-temporal chaos

Tianli Hu·Shijun Liao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Due to the butterfly-effect, computer-generated chaotic simulations often deviate exponentially from the true solution, so that it is very hard to obtain a reliable simulation of chaos in a long-duration time. In this paper, a new strategy of the so-called Clean Numerical Simulation (CNS) in physical space is proposed for spatio-temporal chaos, which is computationally much more efficient than its predecessor (in spectral space). The strategy of the CNS is to reduce both of the truncation and round-off errors to a specified level by implementing high-order algorithms in multiple-precision arithmetic (with sufficient significant digits for all variables and parameters) so as to guarantee that numerical noise is below such a critical level in a temporal interval $t\in[0,T_c]$ that corresponding numerical simulation remains reliable over the whole interval. Without loss of generality, the complex Ginzburg-Landau equation (CGLE) is used to illustrate its validity. As a result, a reliable long-duration numerical simulation of the CGLE is achieved in the whole spatial domain over a long interval of time $t\in[0,3000]$, which is used as a reliable benchmark solution to investigate the influence of numerical noise by comparing it with the corresponding ones given by the 4th-order Runge-Kutta method in double precision (RKwD). Our results demonstrate that the use of double precision in simulations of chaos might lead to huge errors in the prediction of spatio-temporal trajectories and in statistics, not only quantitatively but also qualitatively, particularly in a long interval of time.

Explore related subjects

nlin.CDphysics.comp-ph

Keep this discovery

BibTeXRIS

Tianli Hu, Shijun Liao. 2019-10-26. On the risks of using double precision in numerical simulations of spatio-temporal chaos. https://doi.org/10.1016/j.jcp.2020.109629

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Linear Response Predicts Cusp-Pair Births in Networks with a Localized Cubic

Linear response is cheap to measure; the bistability boundaries it organizes are not. For a passive network with one localized cubic, the driving-point receptance $G$ fixes the period-one cusp set at fundamental-harmonic order: cusps lie on a fixed phase contour of $G$, a tangency of that contour under parameter variation creates a pair, and its curvature separates a gap opening from an isolated loop. For a two-mode absorber the linear prediction locates a benchmark birth coupling to $0.3\%$, and to $0.03\%$ once a third-harmonic correction of scale $|G(3\Omega)/G(\Omega)|$ is included.

nlin.CD↗
arXiv2026

Dynamics Creation through Neural Dynamical Transfer Learning

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed Neural Dynamical Transfer Learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

nlin.CD↗
arXiv2026

The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincar\'e Sections

A Poincar\'e section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the R\"ossler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

nlin.CD↗