Duck trajectories in a thermal explosion problem
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.
scientific article; zbMATH DE number 148538