Embedding Theorems for Maps and Normal Forms for the Slow-fast Systems

Thursday, October 30, 1997 - 11:00am - 12:00pm
Keller 3-180
Yulij Ilyashenko (Cornell University)
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.