Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1908.09683

Massive evaluation and analysis of Poincar\'e recurrences on grids of initial data: a tool to map chaotic diffusion

Ivan I. Shevchenko·Guillaume Rollin·Alexander V. Melnikov·José Lages
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a novel numerical method aimed to characterize global behaviour, in particular chaotic diffusion, in dynamical systems. It is based on an analysis of the Poincar\'e recurrence statistics on massive grids of initial data or values of parameters. We concentrate on Hamiltonian systems, featuring the method separately for the cases of bounded and non-bounded phase spaces. The embodiments of the method in each of the cases are specific. We compare the performances of the proposed Poincar\'e recurrence method (PRM) and the custom Lyapunov exponent (LE) methods and show that they expose the global dynamics almost identically. However, a major advantage of the new method over the known global numerical tools, such as LE, FLI, MEGNO, and FA, is that it allows one to construct, in some approximation, charts of local diffusion timescales. Moreover, it is algorithmically simple and straightforward to apply.

Explore related subjects

nlin.CDastro-ph.EPastro-ph.IM

Keep this discovery

BibTeXRIS

Ivan I. Shevchenko, Guillaume Rollin, Alexander V. Melnikov, José Lages. 2019-08-26. Massive evaluation and analysis of Poincar\'e recurrences on grids of initial data: a tool to map chaotic diffusion. https://doi.org/10.1016/j.cpc.2019.106868

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↗