484 research outputs found
Heterodimensional tangencies on cycles leading to strange attractors
In this paper, we study heterodimensional cycles of two-parameter families of
3-dimensional diffeomorphisms some element of which contains nondegenerate
heterodimensional tangencies of the stable and unstable manifolds of two saddle
points with different indexes, and prove that such diffeomorphisms can be well
approximated by another element which has a quadratic homoclinic tangency
associated to one of these saddle points. Moreover, it is shown that the
tangency unfolds generically with respect to the family. This result together
with some theorem in Viana, we detect strange attractors appeared arbitrarily
close to the original element with the heterodimensional cycle.Comment: 16 pages, 9 figures. to appear in Discrete Conti. Dynam. Sy
Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation
We study bifurcations of a three-dimensional diffeomorphism, , that has
a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers
(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where
and . We show that in a
three-parameter family, g_{\eps}, of diffeomorphisms close to , there
exist infinitely many open regions near \eps =0 where the corresponding
normal form of the first return map to a neighborhood of a homoclinic point is
a three-dimensional H\'enon-like map. This map possesses, in some parameter
regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that
this homoclinic bifurcation leads to a strange attractor. We also discuss the
place that these three-dimensional H\'enon maps occupy in the class of
quadratic volume-preserving diffeomorphisms.Comment: laTeX, 25 pages, 6 eps figure
Infinitely Many Stochastically Stable Attractors
Let f be a diffeomorphism of a compact finite dimensional boundaryless
manifold M exhibiting infinitely many coexisting attractors. Assume that each
attractor supports a stochastically stable probability measure and that the
union of the basins of attraction of each attractor covers Lebesgue almost all
points of M. We prove that the time averages of almost all orbits under random
perturbations are given by a finite number of probability measures. Moreover
these probability measures are close to the probability measures supported by
the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure
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