Stability properties of a class of attractors

Abstract:

Let A be an attractor of an analytical dynamical system defined in ${R^n} \times R$. The class of attractors considered in this paper consists of those A which remain stable as invariant subsets of the complex extension of the flow to ${C^n} \times R$. If A is a critical point or a closed orbit, these are the elementary or generic attractors. It is shown that such an A is always a submanifold of ${R^n}$ and that there exists a Lie group acting on A and containing the given flow as a one parameter dense subgroup; as a consequence, some necessary and sufficient conditions for an analytical dynamical system to have an attracting generic periodic motion are given. It is also shown that for any flow ${C^1}$-close to the given one, there is a unique retraction of a neighbourhood of A onto a submanifold of ${R^n}$ homeomorphic to A that commutes with the flow.