235 research outputs found
Canonical Melnikov theory for diffeomorphisms
We study perturbations of diffeomorphisms that have a saddle connection
between a pair of normally hyperbolic invariant manifolds. We develop a
first-order deformation calculus for invariant manifolds and show that a
generalized Melnikov function or Melnikov displacement can be written in a
canonical way. This function is defined to be a section of the normal bundle of
the saddle connection.
We show how our definition reproduces the classical methods of Poincar\'{e}
and Melnikov and specializes to methods previously used for exact symplectic
and volume-preserving maps. We use the method to detect the transverse
intersection of stable and unstable manifolds and relate this intersection to
the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure
Star fows and multisingular hyperbolicity
A vector field X is called a star flow if every periodic orbit, of any vector
field C1-close to X, is hyperbolic. It is known that the chain recurrence
classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or
singular hyperbolic (see [MPP] for 3-manifolds and [GLW] on 4-manifolds). As it
is defined, the notion of singular hyperbolicity forces the singularities in
the same class to have the same index. However, in higher dimension (i.e ) \cite{BdL} shows that singularities of different indices may be robustly in
the same chain recurrence class of a star flow. Therefore the usual notion of
singular hyperbolicity is not enough for characterizing the star flows.
We present a form of hyperbolicity (called multi-singular hyperbolic) which
makes compatible the hyperbolic structure of regular orbits together with the
one of singularities even if they have different indices. We show that
multisingular hyperbolicity implies that the flow is star, and conversely,
there is a C1-open and dense subset of the an open set of star flows which are
multisingular hyperbolic.
More generally, for most of the hyperbolic structures (dominated splitting,
partial hyperbolicity etc...) well defined on regular orbits, we propose a way
for generalizing it to a compact set containing singular points.Comment: There are new results in section 7 compared with the previous versio
- …