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 Fokker-Planck 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 Fokker-Planck operator. A perturbation in the system results in a perturbation to the Fokker-Planck 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 Hai-Ru Chang)
Spearker : Sebastian Ortega (work with Peter J. Webster, Violeta Toma and Hai-Ru 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 mid-latitude 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/Triple-electron Systems
We discussed the dynamics of collinear Helium near threshold and algebraic decay of triple-electrion surviving probability. Prof. Peijie Wang gave a talk about his research in this area.
Thursday, Feb 12 2015 3:00pm
Title: Series I: Holographic Space-time
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 time-like trajectory in space-time. 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^{d-1}, 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 time-like 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 space-time 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 S-matrix 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 eigen-modes 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: particle-like asymptotic states, meta-stable black hole-like 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 temperature-sensitive 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 spatially-extended dynamics on bounded domains. Additionally, parametric continuation enables the smooth transition from absolute to relative periodic single-spiral solutions. Finally, I will touch on results from the computation of the ``response functions’’ of single- and multi-spiral time-varying solutions.
Thursday, April 16 2015 3:00pm
Spearker: Mohammad Farazmand
Title : Adjoint-based methods for computing equilibrium solutions of partial differential equations
Abstract : I will discuss adjoint-based 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 adjoint-based 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 adjoint-based method on two ex- amples: Kuramoto-Sivashinsky equation and forced two-dimensional 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