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Talk Abstract

Embedding theorems for maps and normal forms for the slow-fast systems

Embedding theorems for maps and normal forms for the slow-fast systems

Slow-fast systems with periodic orbits as attractors for
the corresponding fast systems are called * oscillatory.* The
classical theory of the slow-fast systems mostly deals with those
having singular points as attractors of the fast systems.
We call them * stationary* in order to distinguish from the
oscillatory ones.

The theory of the stationary slow-fast systems in well developed. The theory of
oscillatory ones is in its very beginning. Yet the second theory may be
reduced to the first one through the so-called * embedding theorems. *

The germ of a map at a fixed point is called embeddable if it may be represented as a phase flow transformation of a germ of a vector field, called a generator. Embedding theorems for maps and their families claim that under certain conditions the maps are embeddable; for the maps of smooth families the generator smoothly depends on the parameter. The talk contains new embedding theorems and their applications to the oscillatory slow-fast systems.