Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.1494

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Steven Tomsovic·Arul Lakshminarayan
View original source ↗Read source document ↗Export metadata ↓

Abstract

Some statistical properties of finite-time stability exponents in the standard map can be estimated analytically. The mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants. Whereas the mean carries information about the strength of the interaction, and only indirect information about dynamical correlations, the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength. In particular, the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time-scalings. The average of the stability matrix' inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter. The usual accelerator modes and the small islands created through double saddle node bifurcations, which come halfway between the positions in interaction strength of the usual accelerator modes, are clearly visible in the variance, whose time scaling is capable of detecting the presence of islands as small as 0.01% of the phase space. We study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions. We also discuss the nature of this local approximation in some detail.

Explore related subjects

nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Steven Tomsovic, Arul Lakshminarayan. 2007-06-11. Fluctuations of finite-time stability exponents in the standard map and the detection of small islands. https://doi.org/10.1103/physreve.76.036207

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↗