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Series: CDSNS Colloquium

I'll discuss some work on rigorous computation of invariant
manifolds and computer assisted proof of the existence of transverse
connecting orbits for differential equations. I'm also interested in how
these computations can be used to obtain global topological data, such
as the chain groups and boundary maps of Morse Theory.

Series: CDSNS Colloquium

We shall present a method which establishes existence of normally
hyperbolic invariant manifolds for maps within a specified domain. The
method can be applied in a non-perturbative setting. The required
conditions follow from bounds on the first derivative of the map, and
are verifiable using rigorous numerics. We show how the method can be
applied for a driven logistic map, and also present examples of proofs
of invariant manifolds in the restricted three body problem.

Series: CDSNS Colloquium

This talk is devoted to quasi-periodic Schr\"odinger
operators beyond the Almost Mathieu, with more
general potentials and interactions, considering the
connections between the spectral properties of these
operators and the dynamical properties of the asso-
ciated quasi-periodic linear skew-products. In par-
ticular, we present a Thouless formula and some
consequences of Aubry duality. We illustrate the
results with numerical computations.
This is a join work with Joaquim Puig

Series: CDSNS Colloquium

Consider a generic perturbation of a nearly integrable system
of {\it arbitrary degrees of freedom $n\ge 2$ system}\[H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T,\]with strictly convex $H_0$. Jointly with P.Bernard and K.Zhang we prove existence of orbits $(\th,p)(t)$ exhibiting Arnold diffusion
\[\|p(t)-p(0) \| >l(H_1)>0 \quad \textup{independently of }\eps.\]Action increment is independent of size of perturbation$\eps$, but does depend on a perturbation $\eps H_1$.This establishes a weak form of Arnold diffusion.
The main difficulty in getting rid of $l(H_1)$ is presence of strong double resonances. In this case for $n=2$we prove existence of normally hyperbolic invariant manifolds passing through these double resonances. (joint with P. Bernard and K. Zhang)

Series: CDSNS Colloquium

I will discuss recent work on the stability of linear
equations under
parametric forcing by colored noise. The noises considered are built from
Ornstein-Uhlenbeck vector processes. Stability of the solutions is
determined by the boundedness of their second moments. Our
approach uses the Fokker-Planck equation and the associated PDE
for the marginal moments to determine the growth rate of the
moments. This leads to an eigenvalue problem, which is solved
using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes
into "ladder operators." The results are given in terms of a perturbation
expansion in the size of the noise. We have found very good
agreement between our results and numerical simulations. This is
joint work with L.A. Romero.

Series: CDSNS Colloquium

An important question in circle dynamics is regarding the absolute continuity of aninvariant measure. We will consider orientation preserving circle homeomorphisms withbreak points, that is, maps that are smooth everywhere except for several singular pointsat which the rst derivative has a jump. It is well known that the invariant measuresof sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesguemeasure. But in the case of homeomorphisms with break points the results are quitedierent. We will discuss conjugacies between two circle homeomorphisms with breakpoints.Consider the class of circle homeomorphisms with one break point b and satisfying theKatznelson-Ornsteins smoothness condition i.e. Df is absolutely continuous on [b; b + 1]and D2f 2 Lp(S1; dl); p > 1: We will formulate some results concerning the renormaliza-tion behavior of such circle maps.

Series: CDSNS Colloquium

The discrete Schrodinger operator with Fibonacci potential is a central model in the study of electronic properties of one-dimensional quasicrystals. Certain renormalization procedure allows to reduce many questions on specral properties of this operator to the questions on dynamical properties of a so called trace map. It turnes out that the trace map is hyperbolic, and its measure of maximal entropy is directly related to the integrated density of states of the Fibonacci Hamiltonian. In particular, this provides the first example of an ergodic family of Schrodinger operators with singular density of states measure for which exact dimensionality can be shown. This is a joint work with D. Damanik.

Series: CDSNS Colloquium

With recent advances in experimental imaging, computational methods,
and dynamics insights it is now possible to start charting out the
terra incognita explored by turbulence in strongly nonlinear classical
field theories, such as fluid flows. In presence of continuous
symmetries these solutions sweep out 2- and higher-dimensional
manifolds (group orbits) of physically equivalent states,
interconnected by a web of still higher-dimensional stable/unstable
manifolds, all embedded in the PDE infinite-dimensional state spaces.
In order to chart such invariant manifolds, one must first quotient the
symmetries, i.e. replace the dynamics on M by an equivalent, symmetry
reduced flow on M/G, in which each group orbit of symmetry-related
states is replaced by a single representative.Happy news: The
problem has been solved often, first by Jacobi (1846), then by Hilbert
and Weyl (1921), then by Cartan (1924), then by [...], then in this
week's arXiv [...]. Turns out, it's not as easy as it looks.Still,
every unhappy family is unhappy in its own way: The Hilbert's solution
(invariant polynomial bases) is useless. The way we do this in quantum
field theory (gauge fixing) is not right either. Currently only the
"method of slices" does the job: it slices the state space by a set of
hyperplanes in such a way that each group orbit manifold of
symmetry-equivalent points is represented by a single point, but as
slices are only local, tangent charts, an atlas comprised from a set of
charts is needed to capture the flow globally. Lots of work and not a
pretty sight (Nature does not like symmetries), but one is rewarded by
much deeper insights into turbulent dynamics; without this atlas you
will not get anywhere.This is not a fluid dynamics talk. If you
care about atomic, nuclear or celestial physics, general relativity or
quantum field theory you might be interested and perhaps help us do
this better.You can take part in this seminar from wherever you are by clicking onevo.caltech.edu/evoNext/koala.jnlp?meeting=M2MvMB2M2IDsDs9I9lDM92

Series: CDSNS Colloquium

Hyperbolic actions of Z^k and R^k arise naturally in algebraic
and geometric context. Algebraic examples include actions by
commuting automorphisms of tori or nilmanifolds and, more generally,
affine and homogeneous actions on cosets of Lie groups. In contrast
to hyperbolic actions of Z and R, i.e. Anosov diffeomorphisms and
flows, higher rank actions exhibit remarkable rigidity properties,
such as scarcity of invariant measures and smooth conjugacy to
a small perturbation. I will give an overview of results in this area
and discuss recent progress.

Series: CDSNS Colloquium

A linear cocycle over a diffeomorphism f of a manifold M is
an automorphism of a vector bundle over M that projects to f.
An important example is given by the differential Df or its
restriction to an invariant sub-bundle. We consider a Holder
continuous linear cocycle over a hyperbolic system and explore
what conclusions can be made based on its properties at the
periodic points of f. In particular, we obtain criteria for
a cocycle to be isometric or conformal and discuss applications
and further developments.