150 research outputs found
A (short) survey on Dominated Splitting
We present here the concept of Dominated Splitting and give an account of
some important results on its dynamics.Comment: 19 page
Global bifurcations close to symmetry
Heteroclinic cycles involving two saddle-foci, where the saddle-foci share
both invariant manifolds, occur persistently in some symmetric differential
equations on the 3-dimensional sphere. We analyse the dynamics around this type
of cycle in the case when trajectories near the two equilibria turn in the same
direction around a 1-dimensional connection - the saddle-foci have the same
chirality. When part of the symmetry is broken, the 2-dimensional invariant
manifolds intersect transversely creating a heteroclinic network of Bykov
cycles.
We show that the proximity of symmetry creates heteroclinic tangencies that
coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies -
trajectories that follow the original cycle n times around before they arrive
at the other node. Each n-pulse heteroclinic tangency is accumulated by a
sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes
defined on an infinite set of disjoint strips, where the first return map is
hyperbolic. We also show how, as the system approaches full symmetry, the
suspended horseshoes are destroyed, creating regions with infinitely many
attracting periodic solutions
Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems
We prove a conjecture of J. Palis: any diffeomorphism of a compact manifold
can be C1-approximated by a Morse-Smale diffeomorphism or by a diffeomorphism
having a transverse homoclinic intersection.
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Cr'eation d'intersection homoclines : un mod`ele pour la dynamique centrale
des syst`emes partiellement hyperboliques.
Nous montrons une conjecture de J. Palis : tout diff'eomorphisme d'une
vari'et'e compacte peut ^etre approch'e en topologie C1 par un diff'eomorphisme
Morse-Smale ou par un diff'eomorphisme ayant une intersection homocline
transverse
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
Partial Hyperbolicity and Homoclinic Tangencies
We show that any diffeomorphism of a compact manifold can be C1 approximated
by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms
having a partial hyperbolic structure
Dense heteroclinic tangencies near a Bykov cycle
This article presents a mechanism for the coexistence of hyperbolic and
non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where
trajectories turn in opposite directions near the two nodes --- we say that the
nodes have different chirality. We show that in the set of vector fields
defined on a three-dimensional manifold, there is a class where tangencies of
the invariant manifolds of two hyperbolic saddle-foci occur densely. The class
is defined by the presence of the Bykov cycle, and by a condition on the
parameters that determine the linear part of the vector field at the
equilibria. This has important consequences: the global dynamics is
persistently dominated by heteroclinic tangencies and by Newhouse phenomena,
coexisting with hyperbolic dynamics arising from transversality. The
coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic
and non-hyperbolic dynamics, in contrast with the case where the nodes have the
same chirality.
We illustrate our theory with an explicit example where tangencies arise in
the unfolding of a symmetric vector field on the three-dimensional sphere
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