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

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.

### MSC:

 3.4e+16 Singular perturbations for ordinary differential equations
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