Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1003.2579

On stability of rolls near the onset of convection in a layer with stress-free boundaries

Olga Podvigina
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider a classical problem of linear stability of convective rolls in a plane layer with stress-free horizontal boundaries near the onset of convection. The problem has been studied by a number of authors, who have shown that rolls of wave number $k$ are unstable with respect to perturbations of different types, if some inequalities relating $k$ and the Rayleigh number $R$ are satisfied. The perturbations involve a large-scale mode. Certain asymptotic dependencies between wave numbers of the mode and overcriticality are always assumed in the available proofs of instability. We analyse the stability analytically following the approach of Podvigina (2008) without making a priori assumptions concerning asymptotic relations between small parameters characterising the problem. Instability of rolls to short-scale modes is also considered. Therefore, our analytical results on stability to space-periodic perturbations are exhaustive; they allow to identify the areas in the $(k,R)$ plane, where convective rolls are stable near the onset. The analytical results are compared with numerical solutions to the eigenvalue problem determining stability of rolls.

Explore related subjects

nlin.CDphysics.flu-dyn

Keep this discovery

BibTeXRIS

Olga Podvigina. 2010-03-12. On stability of rolls near the onset of convection in a layer with stress-free boundaries. https://doi.org/10.1080/03091921003603025

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↗