zbMATH — the first resource for mathematics

A generalization of Tichonov theorem. (English) Zbl 0594.34012
Consider the system \[ (1)\quad \frac{dx}{dt}=f(x,y),\quad \epsilon \frac{dy}{dt}=g(x,y), \] \(\epsilon \ll 1\) being a small parameter and \(x\in R^ m\), \(y\in R^ n\). Such systems were first studied by Tichonov. He supposed that a \(C^ 1\)-solution \(y=\phi (x)\) of the algebraic equation \(g(x,y)=0\) is such that for each \(x\in \Omega \subset R^ m\), \(\phi\) (x) represents the one point attractor of the equation (2) \(dy/dt=g(x,y),\) where x is considered as a parameter. Then, provided some technical assumptions are fulfilled, the solutions of the system (1) converge, for \(\epsilon\) \(\to 0\), to the solutions of the system \(dx/dt=f(x,y),\) \(g(x,y)=0\). If the solution \(y=\phi (x)\) of the equation \(g(x,y)=0\) is not a one-point attractor for some x, the Tichonov theorem cannot be used. This may be the case when solutions of the equation (2) are attracted by more complicated attractors. For this and for some of the more general cases, a theorem is given which for the slow and fast components of solutions gives an arbitrarily close approximation provided \(\epsilon >0\) is small enough.
Reviewer: P.N.Bajaj
34A34 Nonlinear ordinary differential equations and systems