Singularly perturbed ordinary differential equations with nonautonomous fast dynamics. (English) Zbl 0937.34044

Artstein, Zvi

J. Dyn. Differ. Equations 11, No. 2, 297-318 (1999).

The author considers the Cauchy problem \[ \begin{alignedat}{2} \frac{dx}{dt}&=f(x,y),\qquad& x(0)&=x_0, \\ \varepsilon\frac{dy}{dt}&=g(x,y,\frac{t}{\varepsilon}),\qquad& y(0)&=y_0, \end{alignedat} \tag{1} \] with \(x\in {\mathbb{R}}^n\), \(y\in {\mathbb{R}}^m\) and \(t\in[0,1]\). In the Levinson-Tichonov theory, a basic assumption is that solutions to \[ \frac{dy}{d\tau}=g(x,y,\tau),\quad y(\tau_0)=y_0, \tag{2} \] converge to an asymptotically stable point \(y=y(x)\). The aim of this paper is to investigate cases where such a property does not hold. The idea is to embed the nonautonomous dynamics in a skew-product flow. Here, solutions to (2) are supposed to be attracted by some compact set \(K(x)\subset {\mathbb{R}}^m\). As a consequence, solutions \((x_{\varepsilon}(t),y_{\varepsilon}(t))\) to (1) are such that every sequence \((\varepsilon_k)_k\) has a subsequence \((\varepsilon_{k_i})_{k_i}\) such that \(x_{\varepsilon_{k_i}}(t)\) converges uniformly on \([0,1]\) to a solution to a differential inclusion \[ \frac{dx}{dt}\in F(x),\quad x(0)=x_0. \] Further, \(y_{\varepsilon_{k_i}}(t)\) converges statistically to a well-defined set of measures. The main drawbacks of such results are that they neither detect the boundary layers nor give expansions in term of the parameter \(\varepsilon\). The author particularizes however his approach to an analog of the Levinson-Tichonov theorem where \(x_{\varepsilon}(t)\) converges uniformly on \([0,1]\) to a solution of a reduced problem and \(y_{\varepsilon}(t)\) converges on \(]0,1]\). At last, an example of application is worked out.

Reviewer: P.Habets (Louvain-La-Neuve)

MSC:

34E15

Singular perturbations for ordinary differential equations

Keywords:

singular perturbations; time-varying equations; skew-product flows; limiting equations; invariant measures