6 research outputs found
Some remarks on the geometry of the Standard Map
We define and compute hyperbolic coordinates and associated foliations which
provide a new way to describe the geometry of the standard map. We also
identify a uniformly hyperbolic region and a complementary 'critical' region
containing a smooth curve of tangencies between certain canonical 'stable'
foliations.Comment: 25 pages, 11 figure
Correlation decay and large deviations for mixed systems
We consider low--dimensional dynamical systems with a mixed phase space and
discuss the typical appearance of slow, polynomial decay of correlations: in
particular we emphasize how this mixing rate is related to large deviations
properties.Comment: 6 pages, 2 figures, submitted to publicatio
Oseledets' Splitting of Standard-like Maps
For the class of differentiable maps of the plane and, in particular, for
standard-like maps (McMillan form), a simple relation is shown between the
directions of the local invariant manifolds of a generic point and its
contribution to the finite-time Lyapunov exponents (FTLE) of the associated
orbit. By computing also the point-wise curvature of the manifolds, we produce
a comparative study between local Lyapunov exponent, manifold's curvature and
splitting angle between stable/unstable manifolds. Interestingly, the analysis
of the Chirikov-Taylor standard map suggests that the positive contributions to
the FTLE average mostly come from points of the orbit where the structure of
the manifolds is locally hyperbolic: where the manifolds are flat and
transversal, the one-step exponent is predominantly positive and large; this
behaviour is intended in a purely statistical sense, since it exhibits large
deviations. Such phenomenon can be understood by analytic arguments which, as a
by-product, also suggest an explicit way to point-wise approximate the
splitting.Comment: 17 pages, 11 figure
On stochastic sea of the standard map
Consider a generic one-parameter unfolding of a homoclinic tangency of an
area preserving surface diffeomorphism. We show that for many parameters
(residual subset in an open set approaching the critical value) the
corresponding diffeomorphism has a transitive invariant set of full
Hausdorff dimension. The set is a topological limit of hyperbolic sets
and is accumulated by elliptic islands.
As an application we prove that stochastic sea of the standard map has full
Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure