Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis

Abstract

We consider one parameter families of vector fields depending on a parameter E such that for epsilon = 0 the system becomes a rotation of R-2 x R-n around {0} x R-n and such that for epsilon > 0 the origin is a hyperbolic singular point of saddle type with, say, attraction in the rotation plane and expansion in the complementary space. We look for a local subcenter invariant manifold extending the stable manifolds to epsilon = 0. Afterwards the analogous case for maps is considered. In contrast with the previous case the arithmetic properties of the angle of rotation play an important role. (c) 2005 Elsevier Inc. All rights reserved.