
wet & wild study group


dynamical systems and fluid dynamics
Summer 2015
 Friday, June 26 2015 10:00am

Title: Perturbation Theory for Noisy Chaotic Systems
Spearker: Jeffrey Heninger
Abstract:
The best description of the long term behavior of a chaotic system is given by the leading eigenfunction of the FokkerPlanck operator. The escape multiplier for the system is the corresponding eigenvalue. Suppose that we have found this eigenfunction and eigenvalue for some combination of deterministic system and noise. Now add a perturbation to the deterministic system, keeping the noise the same. Our challenge is to find out the first order correction to the leading eigenvalue and eigenfunction using only information found the unperturbed FokkerPlanck operator.
A perturbation in the system results in a perturbation to the FokkerPlanck operator. The procedure for finding first order corrections to the eigenvalues and eigenfunctions of an operator can be found in any undergraduate book on quantum mechanics. The only complication is that the normalization is different. We compare the results from the perturbation theory to the results using a direct calculation of the transfer matrix for several perturbations to the Lozi map. The results agree for the perturbed eigenvalues. There is less agreement for the eigenfunctions. This calculation requires multiple unperturbed eigenfunctions. All of them have support on the unperturbed attractor. Perturbation theory fails to find the eigenfunction whenever the perturbed attractor curves off of the support of the original attractor.
 Monday, June 22 2015 3:00pm

Title : Sebastian Ortega (work with Peter J. Webster, Violeta Toma and HaiRu Chang)
Spearker : Sebastian Ortega (work with Peter J. Webster, Violeta Toma and HaiRu Chang)
Abstract :
During summer, the upper level circulation over South Asia is dominated by the monsoon anticyclone. The anticyclone extends from Northeast Africa all the way to the West Pacific Ocean, and it is evident in the upper atmospheric Potential Vorticity (PV) field as a local minimum centered over the Tibetan Plateau. The anticyclone is dynamically unstable, but constantly forced by diabatic heating (convection), and upper level biweekly oscillations seem to be a mechanism by which it dissipates.
These upper level biweekly oscillations follow a characteristic sequence of events that are made clear by studying their associated PV. They are first identified as midlatitude Rossby waves traveling over Asia and breaking over the Pacific Ocean. And later as positive PV anomalies, associated with the breaking Rossby waves, that travel westward over South Asia. Moreover, these oscillations are quite recurrent, and seem to be a persistent feature of the upper troposphere; observed as long as the diabatic forcing is sufficiently strong. More importantly, the oscillations might be related with lower level weather systems that are known to bring copious amount of rain over South Asia.
We define a simple index to study these oscillations, make composite analysis of them to reveal their averaged structure, and show their relation with lower level processes. We show that positive upper level PV anomalies are related to suppress convection immediately bellow. And, in particular, we show that the upper level biweekly oscillations can be linked to weather phenomena such as lower level biweekly oscillations and monsoon depressions.
Additionally, we show preliminary results of a simple shallow water model that seem to capture the essence of the upper level biweekly oscillations. The model simulates the summer time diabatic forcing over the Tibetan Plateau by relaxing its geopotential height to an equilibrium profile, and oscillation arise due to the generation of dynamical instabilities. Interestingly, the model results are quite recurrent, and seem to set the stage for a search of periodic orbits that might help understand the long term statistics associated with these oscillations.
Spring 2015
 Thursday, Jan 15 2015 3:00pm

Burak Budanur
Topics on Dynamic Days US 2015 Conference: Part I
 Thursday, Jan 23 2015 3:00pm

Burak Budanur
Topics on Dynamic Days US 2015 Conference: Part II
 Thursday, Feb 5 2015 3:00pm

Prof. Peijie Wang
Nonliear Dynamics of Double/Tripleelectron Systems
We discussed the dynamics of collinear Helium near threshold and
algebraic decay of tripleelectrion surviving probability.
Prof. Peijie Wang gave a talk about his research in
this area.
 Thursday, Feb 12 2015 3:00pm

Title: Series I: Holographic Spacetime
Speaker: Prof. Tom Banks
Abstract :
I propose a
formalism for the quantum theory of gravitation based on an infinite set
of quantum systems, each of which describes evolution along a particular
timelike trajectory in spacetime. The proper time along all
trajectories is always infinite, either [ \infty , \infty ] or [0,
\infty] and the Hilbert space dimension is related to the maximal area
holographic screen along that trajectory, which might be finite or
infinite. The cosmological constant is fixed by the finite dimension of
this space, if the c.c. is positive, by the asymptotic formula A \sim
t^{d1}, if it is zero. If the c.c. is negative, the area goes to
infinity in finite proper time and the conformal boundary of space time
contains an infinite timelike direction. The theory of propagation
along this direction is a quantum field theory. I won't talk much about
this case, which is the subject of what's known as AdS/CFT. The Hilbert
space in is written as a nested tensor product of Hilbert spaces H_n ,
such that each contains the proceeding as a tensor factor. Each
represents the degrees of freedom accessible in a nested sequence of
causal diamonds. Causality requires the Hamiltonian to be time
dependent such that in some time interval the evolution factors into
independent evolutions inside and outside of a given causal diamond.
The quantum systems along different trajectories are related in the
following way: using Jacobson's idea, we take a discrete set of
trajectories in the spacetime we are trying to model, and look at the
overlaps of the causal diamonds of different trajectories. We must
identify at each time a tensor subfactor of the Hilbert space along one
trajectory with a tensor subfactor of each of the others, and insist the
the density matrices in these subfactors be unitarily equivalent,
according to the evolution along each trajectory. This infinite set of
conditions can actually be solved in certain models, and its qualitative
implications worked out in a variety of others.
 Thursday, Feb 19 2015 3:00pm

Course 1 group members
Group members for the online course discussed some problems
in the homework set including stability of relative equilibrium,
symbolic dynamics, multishooting method and etc.
 Thursday, Mar 26 2015 3:00pm

Title: Series II: The Variables of Quantum Gravity and Some Models
Speaker: Prof. Tom Banks
Abstract :
In the
scattering theory of massless particles in Minkowski space one
encounters the possibility that a finite amount of energy can be emitted
or inserted, in such a way that the energy density at every point of the
holographic screen at infinity vanishes. In four dimensions this leads
to infrared divergences in the conventional definition of the Smatrix
because no process can occur with zero probability for such
emission/absorption. In any dimension, such processes must be included
to preserve unitarity. I propose a definition of scattering in terms of
a mapping between representations of charge density algebras on past and
future infinity, representing flows of quantum numbers into and out of
the holoscreen at infinity. I'll argue that the currents MUST include
spinor currents. On a finite holographic screen we simply restrict the
variables to a finite set of eigenmodes of the Dirac operator, in a
way that implements the Covariant Entropy Principle, in order to
discover the full set of variables of QG. I'll then propose a set of
simple model Hamiltonians for these finite screen variables , all of
which have many of the properties one would want from a theory of
quantum gravity: particlelike asymptotic states, metastable black
holelike intermediate states, long distance interactions scaling with
energy and impact parameter like the Newtonian force, Feynman like
diagram description of interactions which do not involve black hole
formation. Most of them do not satisfy all the consistency conditions
of Abstract I, and will not lead to a Lorentz invariant S matrix, but
there are enough free parameters that one may hope to achieve this as well.
 Thursday, April 02 2015 3:00pm

Michael Dimitriyev
Graduate student Michael Dimitriyev talks about his research
on deswelling and buckling of temperaturesensitive
hydrogel tori.
 Thursday, April 09 2015 3:00pm

Spearker: Christopher Marcotte
Title : Local Euclidean symmetry in a simple model of
atrial fibrillation: a tale of infinite spirals
Abstract : In this talk I will attempt to concisely present results which
explore the phenomenon of local symmetry in a simple model of atrial
fibrillation. In particular, this phenomenon permits the existence of
relative periodic solutions to spatiallyextended dynamics on bounded domains.
Additionally, parametric continuation enables the smooth transition from
absolute to relative periodic singlespiral solutions. Finally, I will touch on
results from the computation of the ``response functions’’ of single and
multispiral timevarying solutions.
 Thursday, April 16 2015 3:00pm

Spearker: Mohammad Farazmand
Title : Adjointbased methods for computing equilibrium
solutions of partial differential equations
Abstract : I will discuss adjointbased methods for computing
the equilibrium solutions of evolution equations of the form
du/dt = F(u), where F is a nonlinear differential operator and
u(x, t) is a function of space x ∈ Ω ⊂ R+. The equilibrium solutions
u = u(x) of such PDEs are often found t ∈ R using some variant of Newton
iterations to solve the nonlinear set of equations F(u) = 0. Such Newton
iterations are computationally expensive and often do not converge unless
a very good initial guess is provided.
The adjointbased method, on the other hand, seeks the equilibria u by
signed in such a way that (1) Its equilibria include all equilibria of the original
solving an adjoing PDE of the from ∂tu PDE ∂tu = F(u) and (2) The equilibria
of the adjoint PDE are stable. The adjoint operator G can be explicitly
described for a significant set of PDEs and boundary conditions. Due to its
stability, the numerical integration of the adjoint equation is significantly cheaper
than the original PDE.
I will illustrate the performance of the adjointbased method on two ex
amples: KuramotoSivashinsky equation and forced twodimensional Navier–
Stokes equations.
 
Previous schedules
Wet & Wild Fall 2013
Wet & Wild 2014
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meets Thursdays at 3:00pm
Howey W505 conference room  all are welcome to join
organizer:
Xiong Ding, xding (snail) gatech.edu