Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.26480

Scrambling at the genesis of chaos

Thomas R. Michel·Mathias Steinhuber·Juan Diego Urbina·Peter Schlagheck
View original source ↗Read source document ↗Export metadata ↓

Abstract

The presence of chaos in classical Hamiltonian systems is witnessed by its maximal Lyapunov exponent, that quantifies the instability of motion through the exponential growth of indicators such as the trace of the stability matrix or the out-of-time-ordered correlator. On the other hand, integrable dynamics near unstable fixed points, which are in turn characterized by a stability exponent, can also induce such exponential growth. Following the paradigm of integrability-breaking as driven by nonlinear resonances that hallmarks the genesis of chaos, the integrability-chaos transition is universally described by a periodic perturbation applied to a generic pendulum. Remarkably, this means that within the corresponding separatrix dynamics, which is an unavoidable a consequence of the resonance scenario, both instability exponents must play a role as both dynamical regimes coexist. We report here the universality of the transition from instability to Lyapunov exponents, thus completing the resonance scenario at the level of indicators based on exponential growth. To achieve this goal we obtain an analytical expression for the time evolution near separatrices, which enables us to derive an analytical expression for the exponent that characterises chaos and its transition from local instability to global chaos. We support our claim for the universality of this mechanism by studying two paradigmatic examples of the integrability-to-chaos transition, namely the kicked rotor and the driven pendulum.

Explore related subjects

nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Thomas R. Michel, Mathias Steinhuber, Juan Diego Urbina, Peter Schlagheck. 2026-03-27. Scrambling at the genesis of chaos. https://arxiv.org/abs/2603.26480

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↗