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Nonautonomous attractors of skew-product flows with digitized driving systems. (English) Zbl 1005.34050

The authors study the semicontinuity and continuity properties of pullback attractors for nonautonomous differential equations \[ x' = f(t,x), \qquad x \in \mathbb{R},\quad t \in \mathbb{R}, \] under a digitization procedure. These pullback attractors exist e.g.under certain dissipativity conditions. The problem is formulated using the language of skew-product flows, in particular the Bebutov flow [G. R. Sell, Topological dynamics and ordinary differential equations. London etc.: van Nostrand Reinhold Company (1971; Zbl 0212.29202)]. By digitizing the authors mean the following: the time-axis is decomposed into half-open intervals of lengths, say, between \(\delta / 2\) and \(\delta\) for each \(\delta > 0\). On each such interval, the time-varying vector field \(f(t,x)\) is replaced by an autonomous vector field \(\overline{f}(x)\) (which usually depends on the particular interval). For example, one may choose \(\overline{f}(x)\) to be the time-average of \(f(t,x)\) over the given (or previous) interval, or as some particular value \(f(t_\star,x)\), or in some other way as well.
The authors prove that the pullback attractor of the digitized system converges upper semicontinuously to the pullback attractor of the original system as \(\delta \to 0\), with respect to the Hausdorff metric on compact subsets of \(\mathbb{R}^d\). Under additional assumptions such as existence of an exponential dichotomy for the linearization and uniform contractivity of the nonlinearity, the authors prove continuous convergence to the pullback attractor of the original system as \(\delta \to 0\), with respect to the Hausdorff metric on compact subsets of \(\mathbb{R}^d\). An example of a quasi-periodic vector field is given to illustrate the results.

MSC:

34D35 Stability of manifolds of solutions to ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
37B55 Topological dynamics of nonautonomous systems

Citations:

Zbl 0212.29202
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