On the stability of smooth dynamical systems and diffeomorphisms

Andrzej Zajtz (KrakÓw)

Abstract

General methods of studying global stability problems of smooth dynamical systems and diffeomorphisms are presented.

In particular, one proves that any complete C^{∞} vector field X in a Hilbert space E satisfying ⟨X(x), v⟩ ≥ δ > 0 for some constant field v and all x ∈ E (for instance, if ∥X −v∥ ≤ \frac{∥v∥}{2}) is globally rectifiable to v by a C^{∞} diffeomorphism of E. Thus any nonzero constant vector field in E is smoothly stable in a 0-order neighborhood.

Similarly one obtains the global stability (in a 1st-order neighborhood) of expansive non-resonant linear systems.

Institute of Mathematics, Pedagogical Academy ul. Podchor a żych 2, 30-084 Kraków POLAND e-mail: smzajtz@cyf-kr.edu.pl