Seminars and Colloquia by Series

Spectral Submanifolds and Exact Model Reduction for Nonlinear Beam Dynamics

Series
CDSNS Colloquium
Time
Friday, January 13, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Florian Kogelbauer – ETH (Zurich)
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.

The nonlinear Schroedinger equation and the evolution of wave packets in nonlinear dispersive equations

Series
CDSNS Colloquium
Time
Wednesday, November 30, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Eugene Wayne – Boston University
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.

Hamiltonians and normal forms for water waves

Series
CDSNS Colloquium
Time
Wednesday, November 16, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
006
Speaker
Prof. Walter Craig – McMaster University
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.

Supperdiusion constants for certain nonuniformly hyperbolic systems

Series
CDSNS Colloquium
Time
Monday, October 24, 2016 - 11:06 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hongkun Zhang – U. Mass Amherst
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)

Pairings between periodic orbits in hyperbolic coupled map lattices.

Series
CDSNS Colloquium
Time
Monday, October 17, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Boris Gutkin – Georgia Tech (School of Physics)
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.

Locally integrable non-Liouville analytic geodesic flows on T^2

Series
CDSNS Colloquium
Time
Monday, August 29, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Livia Corsi – Georgia Tech - School of Math
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.

Parameterization of periodic invariant objects for maps

Series
CDSNS Colloquium
Time
Monday, May 2, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
J. Mireles-James – Florida Atlantic Univ.
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.

Rigorous validation of Radially Symmetric Stationary Solutions of PDEs.

Series
CDSNS Colloquium
Time
Monday, May 2, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
C.M. Groothedde – V.U. Amsterdam
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.

Hamiltonian Instability in a Four-Body Problem

Series
CDSNS Colloquium
Time
Monday, April 25, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marian Gidea – Yeshiva Univ.
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

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