Tracking nonautonomous attractors in singularly perturbed systems of ODEs with dependence on the fast time

TL;DR

This paper extends classical Tikhonov theory to nonautonomous slow-fast ODE systems, analyzing the behavior of fast motion through nonautonomous attractors and establishing conditions for their tracking and stability.

Contribution

It introduces new results on tracking nonautonomous attractors in singularly perturbed systems with dependence on fast time, generalizing existing autonomous attractor theory.

Findings

Inflated pullback attractors are crucial under general assumptions.

Continuity of the fiber projection map ensures the relevance of inflated attractors.

Results extend Tikhonov's theorem to nonautonomous systems.

Abstract

New results on the behaviour of the fast motion in slow-fast systems of ODEs with dependence on the fast time are given in terms of tracking of nonautonomous attractors. Under quite general assumptions, including the uniform ultimate boundedness of the solutions of the layer problems, inflated pullback attractors are considered. In general, one cannot disregard the inflated version of the pullback attractor, but it is possible under the continuity of the fiber projection map of the attractor. %In particular this happens when the attractors of the layer problems are copies of the base, which is the counterpart of an asymptotically stable equilibrium point in the autonomous case. The problem of the limit of the solutions of the slow-fast system at each fixed positive value of the slow time is also treated and in this formulation the critical set is given by the union of the fibers of the pullback attractors. The results can be seen as extensions of the classical Tikhonov theorem to the nonautonomous setting.