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| In the classical applications of the geometric singular perturbation theory or of the Tikhonov theorem to dynamical systems with a small parameter, it is required that the limit manifold (quasi steady state) is isolated and attracting in a specific domain. However, often the quasi steady states intersect and switch stability. It is the expected that solutions to such systems with the parameter small enough, having followed the attracting branch of one of the quasi steady states, after passing close to the intersection, will switch stability and start following the attracting branch of the other quasi steady state. It turns out, however, that in many cases the solutions will continue to follow the repelling branch of the first quasi steady state for a considerable time before switching to the new, attracting one. In this talk we shall discuss generalizations and applications of the Butuzov theorem which deals with such problems but only in one dimension. |
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