- You are here:
- GT Home
- Home
- News & Events

Series: CDSNS Colloquium

We use invariant manifold
results on Banach spaces to conclude the existence of spectral
submanifolds (SSMs) in a class of nonlinear, externally forced beam
oscillations .
Reduction of the governing PDE to the SSM provides an exact
low-dimensional model which we compute explicitly. This model captures
the correct asymptotics of the full, infinite-dimensional
dynamics. Our approach is general enough to admit extensions to other
types of continuum vibrations. The model-reduction procedure we employ
also gives guidelines for a mathematically
self-consistent modeling of damping in PDEs describing structural vibrations.

Series: CDSNS Colloquium

The nonlinear Schroedinger
equation (NLS) can be derived as a formal approximating equation for the
evolution of wave packets in a wide array of nonlinear dispersive PDE’s
including the propagation of waves on the surface of an inviscid
fluid. In
this talk I will describe recent work that justifies this approximation
by exploiting analogies with the theory of normal forms for ordinary
differential equations.

Series: CDSNS Colloquium

It was shown by V.E. Zakharov that the equations for water waves can be posed as a Hamiltonian PDE, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations. This latter is joint work with C. Sulem.

Series: CDSNS Colloquium

We investigate deterministic superdiusion in nonuniformly hyperbolic system
models in terms of the convergence of rescaled distributions to the normal
distribution following the abnormal central limit theorem, which differs
from the usual requirement that the mean square displacement grow
asymptotically linearly in time. We obtain an explicit formula for the
superdiffusion constant in terms of the ne structure that originates in the
phase transitions as well as the geometry of the configuration domains of
the systems. Models that satisfy our main assumptions include chaotic
Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply to
other nonuniformly hyperbolic systems with slow correlation decay rates of
order O(1/n)

Series: CDSNS Colloquium

Upon quantization, hyperbolic Hamiltonian systems generically exhibit
universal spectral properties effectively described by Random
Matrix Theory. Semiclassically this remarkable phenomenon can be
attributed to the existence of pairs of classical periodic orbits
with small action differences. So far, however, the scope of this
theory has, by and large, been restricted to single-particle systems.
I will discuss an extension of this program to hyperbolic
coupled map lattices with a large number of sites (i.e., particles).
The crucial ingredient is a two-dimensional symbolic dynamics which
allows an effective representation of periodic orbits and their
pairings. I will illustrate the theory with a specific model of
coupled cat maps, where such a symbolic dynamics can be constructed
explicitly.

Series: CDSNS Colloquium

A metric on the 2-torus T^2 is said to be "Liouville" if in some coordinate system it has the form ds^2 = (F(q_1) + G(q_2)) (dq_1^2 + dq_2^2). Let S^*T^2 be the unit cotangent bundle.A "folklore conjecture" states that if a metric is integrable (i.e. the union of invariant 2-dimensional tori form an open and dens set in S^*T^2) then it is Liouville: l will present a counterexample to this conjecture.Precisely I will show that there exists an analytic, non-separable, mechanical Hamiltonian H(p,q) which is integrable on an open subset U of the energy surface {H=1/2}. Moreover I will show that in {H=1/2}\U it is possible to find hyperbolic behavior, which in turn means that there is no analytic first integral on the whole energy surface.This is a work in progress with V. Kaloshin.

Series: CDSNS Colloquium

The Parameterization Method
is a functional analytic framework for studying invariant manifolds
such as stable/unstable manifolds of periodic orbits and invariant
tori. This talk will focus on numerical applications such as computing
manifolds associated with long periodic orbits, and computing periodic
invariant circles (manifolds consisting of several disjoint circles
mapping one to another, each of which has an iterate conjugate to an
irrational rotation). I will also illustrate how to combine Automatic
Differentiation with the Parameterization Method to simplify problems
with non-polynomial nonlinearities.

Series: CDSNS Colloquium

We shall take a look at computer-aided techniques that can be
used to prove the existence of stationary solutions of radially
symmetric PDEs. These techniques combine existing numerical methods with
functional analytic estimates to provide a computer-assisted proof by
means of the so-named 'radii-polynomial' approach.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

We consider a restricted four-body problem, modeling the dynamics of a
light body (e.g., a moon) near a Jupiter trojan asteroid. We study two
mechanisms of instability. For the first mechanism, we assume that the
orbit of Jupiter is circular, and we investigate the hyperbolic invariant
manifolds associated to periodic orbits around the equilibrium points. The
conclusion is that the light body can undergo chaotic motions inside the
Hill sphere of the trojan, or well outside that region. For the second
mechanism, we consider the perturbative effect due to the eccentricity of
the orbit of Jupiter. The conclusion is that the size of the orbit of the
light body around the trojan can keep increasing, or keep decreasing, or
undergo oscillations. This phenomenon is related to the Arnold Diffusion
problem in Hamiltonian dynamics