Duck trajectories in a thermal explosion problem
type: Item
instance of: scholarly article
title: Duck trajectories in a thermal explosion problem
zbMATH Open document ID: 0767.34040
DOI: 10.1016/0893-9659(92)90002-Q
author: G. N. Gorelov
author: V. A. Sobolev
published in: Applied Mathematics Letters
publication date: 1993-04-01T00:00:00Z
review text: The authors study the system of ODEs (1) \(\varepsilon dx/dt=f(x,y,\alpha)\), \(dy/dt=g(x,y)\), where \(x\), \(y\) are scalar variables, \(\alpha\) is a scalar parameter and \(\varepsilon\) is a small positive parameter. The set of points \(s=\{(x,y): f(x,y,\alpha)=0\}\) on the phase plane of system (1) is called the slow curve. The trajectory of system is called a duck trajectory if it passes infinitely close to the slow curve. The authors discuss the duck trajectories in a thermal explosion problem, as examples the systems (2) \(\varepsilon dx/dt=x^ 2- y^ 2+\alpha\), \(dy/dt=1\) and (3) \(\varepsilon d\vartheta/d=\eta(1- \eta)e^ \vartheta-\alpha\vartheta\), \(d\eta/d\tau=\eta(1-\eta)e^ \vartheta\) are discussed, where \(\alpha=\pm\varepsilon\). The lines \(x=\pm y\) pass along the curve \(x^ 2-y^ 2+\alpha=0\) at an infinitely long distance, notice that the duck trajectory is only the \(x=y\) trajectory for system (2). In the system (3) \(\vartheta\), \(\eta\) and \(\tau\) are dimensionless temperature, where \(\alpha\) is an important parameter, which characterizes the initial state of system, determines the reaction process which leads to an explosion. The authors examine the critical values of a parameter in a thermal explosion problem in the autocatalytic case.
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 80A25
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 148538
zbMATH Keywords: duck trajectory
zbMATH Keywords: thermal explosion
MaRDI profile type: Publication
cites work: Relaxation oscillations including a standard chase on French ducks
cites work: Differential equations with small parameters and relaxation oscillations. Translation from the Russian by F. M. C. Goodspeed
cites work: Integral manifolds and decomposition of singularly perturbed systems
cites work: Non-standard analysis and singular perturbations of ordinary differential equations
full work available at URL: 0893-9659(92)90002-q
OpenAlex ID: W2010340872
description: scientific article; zbMATH DE number 148538