Immediate exchange of stabilities in singularly perturbed systems
The authors study initial value problems for the system of ordinary differential equations \[ \frac {dx}{dt}=f(x,y,t,\varepsilon),\quad \varepsilon \frac {dy}{dt}=g(x,y,t,\varepsilon),\tag{1} \] where \(x\in \mathbb{R}^n\), \(y\in \mathbb{R}^m\) and \(\varepsilon \) is a small positive parameter, which is considered as a perturbation of the differential-algebraic system \[ \frac {dx}{dt}=f(x,y,t,0),\quad 0=g(x,y,t,0). \tag{2} \] In the theory of invariant manifolds, system (2) is usually considered under the assumption that its algebraic part \(g(x,y,t,0)=0\) has an isolated solution \(y=\varphi (x,t)\) on which all eigenvalues of the Jacobian \(g_y(x,\varphi (x,t),t,0)\) are located in the left half plane. Here, the case is considered when \(g(x,y,t,0)=0\) has two solutions \(\varphi _1\) and \(\varphi _2\) which intersect and exchange their stabilities. Then the following question arises: when a solution to (1) which stays near \(\varphi _1\) before the exchange of stability will immediately stay near \(\varphi _2\) after the exchange of stability? A sufficient condition for such a phenomenon is derived by means of the method of upper and lower solutions.
scientific article; zbMATH DE number 1858761