Transport in volume-preserving maps and control of Hamiltonian systems Umesh Vaidya
My research interest is in the area of theoretical and computational aspects
of transport in volume-preserving maps and control of Hamiltonian systems.
Geometrical structures such as invariant manifolds and Smale horseshoe play an
important role in the transport dynamics of volume-preserving maps. In my
research on three-dimensional volume-preserving maps, I have proved KAM type
of result for the persistence of one and two-dimensional invariant manifolds.
These results have interesting application in the emerging technology of
microfluidic devices, where they can be used to build efficient micromixers.
In my work on controllability of Hamiltonian systems, by exploiting the
natural dynamics of the system in particular the ergodic property of the
drift, we have given conditions for controllability which are stronger than
currently existing conditions based on Lie Algebra. The proposed
controllability approach suggested in this work also gives insight for ways of
constructing the control. Apart from this I am also interested in the problem
of control of mixing and control of quantum systems.
Control Theory - Dr. Dmitri Vainchtein
In many two-dimensional fluid flows with coherent structures, vortex
interactions can be reduced to
pairwise interactions of vortices. We are interested in controlling the behavior
of these structures by reducing or enhancing the pairwise interactions.
When the vortices are far apart, the point vortex approximation can be used. The
problem dimension is reduced by averaging over the fast rotation of vortices around the center of
vorticity. We consider strain and source/sink actuation fields. We show that optimal solutions
when only the
average power is bounded consist of Dirac delta functions applied at optimal
phases during the
cycle of vortex rotation. If a bound on the control magnitude is added to the
constraints a
solution is obtained for which control is at the maximum amplitude except
possibly for intervals
during which it is zero. For the case of a source-sink field, we use the
Pontryagin maximum
principle to obtained the optimal solution. Impulsive control arises here
naturally as an optimal
control design once averaging method is used.
For two close vortices, we discuss two methods of controlling the motion of two
elliptical vortex
patches. One way is to use the method of flat coordinates (a brute force
method). Hamiltonian
structure of control field is important to prove controllability and Hamiltonian
structure of
nominal system leads to significant extension of reachable domain. The other
approach is to use
control that is a small perturbation compared with the nominal system. We show
that the optical
control for the second setting is a set of impulses in phase with fast (angle)
variable of the
nominal system. We compare the two approaches and discuss their relative
advantages. We compare
predictions of the model with results of the numerical experiments.
Nonlinear dynamics and control of power grids -
Dr. Yoshihiko Susuki
Power grid is a typical networked and safety-critical system with large
number of components (generation unit, transmission line, load, substation,
etc.) and complex network topology. Power grid has been faced with novel
technical and nontechnical trends: occurrence of 2003 grid blackouts in
North America and Europe, development of distributed power sources such as
solar and wind power generations, necessity of reduction of environmental
burdens in thermal generation units, and deregulation of electricity
markets. These trends require a new framework of analysis and control of
large scale, complex, and hierarchical power grids. The purpose of our
study is to establish a new power grid engineering from viewpoints of
nonlinear dynamics and control. We are currently studying a new concept of
grid instability, called global instability of power grid.
Approximation of ergodic partition of dynamical systems' phase spaces - Marko Budisic
A general dynamical system can exhibit a variety of qualitatively different behaviors depending on the initial phase space point from which it is run. A useful beginning of analysis of a dynamical system could be catalogization of regions on which system has uniform behavior. Decomposition of phase space into invariant sets (if system is started from a point in an invariant set, it never leaves the set) is one possible approach to take. A finest of many possible decompositions of that type is the ergodic decomposition whose elements are ergodic sets (system samples ergodic sets "uniformly" if started from them).
My goal is to produce and theoretically support a computational tool that will efficiently approximate large-scale features of ergodic decomposition for arbitrary, but computable, dynamical systems. Koopman-like operator method is used to compute features in phase space. These features are aggregated into larger structures using a sequence of spectral clustering methods.
[The image shows a coarse decomposition of standard map phase space.]
The Effect of Symmetry on Spatiotemporal Resonances in Coupled Systems -
Bryan Eisenhower
This work focuses on the effect of coupling nonlinear cells primarily focusing on local and global resonance phenomena and how spatial symmetry effects the global response. The term cells loosely describes any inter-connectable dynamical system (e.g. local molecular dynamics, thermoacoustic jet engine dynamics, buildings systems, interaction in human decision making, etc) that once connected act together in some way (good or bad) to achieve a global goal. In many cases the coupling and subsequent global response is inherent and a behavior we have to live with while in other cases, the coupling is a design parameter. The outcome of this work is design tools to enable exploitation of the resonance phenomena when desired, or quenching when it is undesirable.
Averaging of dynamical systems - Thomas John
I'm a Ph.D candidate in Mechanical Engineering and a Masters student
in Mathematics. I am interested in averaging in dynamical systems,
geometrical methods and their application to fluid mechanics in the
context of the Lagrangian averaged Euler and Navier-Stokes (LAE/LANS)
equations. A problem of particular interest is the use of the LAE as
a regularization for the Euler equations and its application to
mollify the singularity in the vortex sheet problem.
Some problems of industrial interest that I have worked on include low
dimensional modeling on a jet-in-crossflow for use in the combustion
chamber of an aircraft engine and a method for the efficient use of
air-conditioning system to expel contaminants that have leaked into a
room. I also like interesting math problems.
Computation and Visualization of Invariant Sets of Dynamical Systems Dr. George Mathew
Computation and Visualiazation of Invariant Sets of Dynamical Systems
is a very important tool in the analysis of chaotic systems. It gives
valuable information about the global dynamics and qualitiative
behaviour of the system without having to solve the underlying
dynamical system explicitly. Here, we attack the problem from the point
of view of the so-called Koopman operator which acts on the space of
observables and apply it to the Standard Map. The figure shows
the dynamics of the computed invariant sets of the Standard Map with
respect to the perturbation parameter.
Visualization of Invariant Sets in Discrete-time Dynamical Systems Zoran Levnajic
My current research is focused on finding computational techniques to study statistical properties of
discrete-time dynamical systems like standard map. With the aid of Ergodic Partition Theory I made
algorithms and codes that are computing quantities of interest like time averages of certain functions
under the dynamics thus giving information on existence and distribution of invariant sets within the
domain of map. Moreover, I am considering harmonic time averages trying to identify periodic subsets
as opposed to ergodic (chaotic) subsets in the map's phase space. The computations are done using
Parallel Fortran 77 while the plots themselves are made with MatLab. Apart the standard map, mentioned
methods are presently being applied to various maps like extended standard map, Froeschle map and are
soon to be applied to a class of physical systems - coupled oscillators, billiards and such. The picture
represents the time average under standard map for a Haar wavelet function.
Resonance phenomena and destruction of adiabatic invariants Dr. Dmitri Vainchtein
We study the effects of singularities such as separatrix crossings and capture
into resonance and
scattering on resonance on near-integrable systems. In the absence of
singularities KAM theory
guarantees almost regular dynamics. However, in the presence of resonances
arbitrarily small
perturbations of a certain kind, domains of chaotic advection arises. We show
that this phenomenon
is a consequence of quasi-random changes in the adiabatic invariant of the
system, which occur as a
streamline crosses the resonance surface of the unperturbed flow. We derive an
asymptotic formula
for the change in the adiabatic invariant due to the passages through the
resonance and describe
the diffusion of the adiabatic invariant due to multiple passages through the
resonance. As a
possible development of this technique, we show how (by adding a even smaller
additional control)
resonance phenomena may result not in a chaotic advection, but a regular
transport.
Graph theoretic analysis and dynamical systems approach on large-scale biological networks - Dr . Lan Yueheng
The advent of Gene-sequencing and high throughput experiments
provides, with unprecendented large collection of data, the opportunity for
the construction and analysis of large-scale biological networks in
cell that involve thousands of different macromolecular species and
biochemical reactions. The concomitant challenge has to be met with
combination of tools from such diverse fields as mathematics, physics,
chemistry, engineering. Here, we apply graph theoretic techniques to
analyze the topological structure of large-scale networks, concentrating on
cyclic components and pipeline connections between them. Our goal is to
identify the functional motifs and explore how they are wired into the
hierarchical modules which are a universal character and the function
realizer of the robust and disparate biological behaviors. The dynamical
systems theory are invoked to simplify the modules and check the
consequence of sequentially piecing together the basic motifs.
Mixing in microchannels - Dr. Frederic Bottausci
We study experimentally the fluidic mixing at the microscale. The active
micromixer is composed of one main channel where the fluids are injected
and three pairs of side channels. Mixing is achieved in a laminar flow by
perturbing the main flow with transverse impinging jets from secondary
side channels. This stretches and folds the layers in the flow stream
causing chaotic advection, thereby increasing mixing. The current mixer is
a silicon-etcheddevice with a glass cover slip anodically bonded on top
to hermetically seal the chip. The main channel is 200 microns wide, 100
microns deep and 1300 microns long. Experiments are performed with either
the first, the two first or all three side channels activated. The flow is
pressure driven in the side channels using a specially developed
oscillating syringe pump and is controlled using a software/hardware
Labview. The working fluids injected in the main channel consist of a
fluorescent aqueous solution and deonized water. The time evolution of
the flow is observed using an epi-fluorescent microscope and a YAG laser.
The flow is characterized by micro-PIV measurements and visualizations.
Mixing is quantified using the Mixing Variance Coefficient function. We
achieve a good mixing of the two fluids when the three side channels are
activated within 0.1 second.
Numerical simulation of the mixing in a microchannel - Caroline Cardonne
I am performing 3D numerical simulations in order to understand and optimize the mixing at the
microscale in an active micromixer. I am using the CFD software Fluent for the computation and Gambit to create the geometry.The micromixer I am working on is composed of one main channel, where the fluids to be mixed are injected,
and of three pairs of side channels. The mixing is achieved in the laminar flow by perturbing the main flow with transverse oscillating flows imparted in the secondary
side channels. This stretches and folds the layers in the flow stream causing chaotic advection, thereby increasing mixing. The main channel is 200 microns wide, 100 microns
deep and 1350 microns long. Simulations were performed with either the first or all three side channels activated. The mixing is quantified using the Mixing Variance Coefficient function.
We achieve a good mixing of the two fluids whith the three side channels activated within 0.1 second.
A Multiscale Measure for Mixing and its Applications - Dr. George Mathew
In spite of a large amount of recent research on the problem of fluid
mixing and its control, there is no consensus on a proper measure for
quantifying mixing. We present a measure of mixing that is based on the
concept of weak convergence and is capable of probing the ``mixedness''
at various scales. This new measure, called the Mix-Norm, resolves the
inability of the scalar variance of the scalar density field to resolve
various stages of contour-level rearrangement by chaotic maps. In
addition, the Mix-Norm succeeds in capturing the efficiency of a mixing
protocol in the context of a particular initial density field, wherein
Lyapunov-exponent based measures fail to do so. The Mix-Norm is a
pseudo-norm for checking weak convergence on the space of scalar
density fields, which turns out to be a critical link in justifying its
validity as a measure for mixing. We demonstrate the utility of the
Mix-Norm by showing how it measures the efficiency of mixing due to
diffusion and to various discrete dynamical systems.
Computation and Visualization of Invariant Sets of Dynamical Systems
Dr. George Mathew
Computation and Visualiazation of Invariant Sets of Dynamical Systems
is a very important tool in the analysis of chaotic systems. It gives
valuable information about the global dynamics and qualitiative
behaviour of the system without having to solve the underlying
dynamical system explicitly. Here, we attack the problem from the point
of view of the so-called Koopman operator which acts on the space of
observables and apply it to the Standard Map. The figure shows
the dynamics of the computed invariant sets of the Standard Map with
respect to the perturbation parameter.
Particles separation and micro mixing - Dr.Frederic Bottausci
We present experiments on dielectrophoretic (DEP) separation and trapping performed in a titanium-based
micro channel linear electrode array. The micro channel was fabricated in collaboration with Y. Zhang from
N. MacDonald's group. The device consists of an array of 24 electrodes sitting on the bottom of 200 microns
wide, 30 microns deep and 6 millimeters long titanium channel. The electrodes are 20 microns wide with a pitch
of 40 microns. The channel is versatile and biocompatible. The device is designed to allow multi-frequency DEP
(p-DEP and n-DEP) in contrast with most of the previous, single-frequency designs.
We experimentally demonstrated the ability to separate fluorescent polystyrene particles based or their size.
More experiments are in process to study the separation for:
- biological samples
- different electrolyte conductivities
- optimization
The device also allows the use of traveling waves to move particles using non-homogeneities in electric-field
phase-driven DEP. The idea is to separate the particle and move them in specific location. The first experiments
are in process.
By tuning the multi-frequency signal, we have also shown the ability of producing strong micro mixing. Once
the particles are trapped, we have shown experimentally and theoretically that a small perturbation can strongly
destabilized the flow.
Polymerase Chain Reaction - Dr. Frederic Bottausci
Polymerase Chain Reaction (PCR) becomes more and more important for the detection of diseases and other
biological study. We are fabricating and testing a new device enabling fast PCR for lab-on-chip application.
Separation of Bioparticles using the travelling wave dielectrophoresis with multiple frequencies - Sophie Loire
Atomic Force Microscope (AFM) is an essential characterization and manipulation tool
in nanoscience and engineering. It's operating method is based on the dynamical behavior
of a microcantilever-sample system. The traditional research on AFM uses a one-mode model.
My goal is to find the effect of higher modes on this nonlinear dynamical system. We analyze
the forced dynamics of a microcantilever where the force on the tip is derived from the Lennard-Jones
potential and thus depends on the displacement of the tip mass. Prior single-mode analysis showed
complex nonlinear dynamics in some regimes. We present numerical and analytical results that confirm
the necessity of inclusion of higher modes for the modeling of the AFM dynamics. Electric fields are
widely used to manipulate microparticles. Among various forms of electric fields we focus on
dielectrophoresis (DEP) ; the motion of a particle due to the interaction between a non-uniform
electric field and its induced dipole moment in the particle. The fields are applied to a suspension
of particles by planar microelectrode structures. We study one particular design, the interdigitated
electrode arrays with either two or four-phase signal. We worked on closed-form solutions of electric
fields, dielectrophoretic forces and time-averaged forces for three cases : first, the case of a
two-phase DEP electrode array with first-order approximate boundary condition, second, the case of
a two-phase DEP electrode array with exact boundary condition and lastly the case of four-phase
traveling wave (twDEP) electrode array with first-order approximate boundary condition. We discuss
those results with a comparison to numerical solutions. One usually uses DEP force to separate two
kinds of particles using the difference in their dielectric properties. We develop a method of
separation with multiple frequency dielectrophoresis which permits to separate bioparticles with
very close dielectric properties while ordinary one-frequency methods fail.
Trapping of nanoparticles and cold neutral atoms - Sophie Loire
I work on modeling the apparition of trapping of nano-sized
particles using a
combination of DEP, fluid flow (such as AC electroosmosis) and Brownian
motion.
I
also study the trapping of cold neutral atoms in an
electro-magnetic ring trap. I investigate the effect of time-dependency
of the electric field on the dynamical behaviors of the particles in
these two configurations and consider the optimization of these methods.
Dynamics of self assembling protein molecules - Gunjan Thakur
There is a considerable interest in the mechanism governing the super
molecular assembly of proteins in biological systems. Recently discovered
families of proteins, called silicatines, are interesting candidates for
the use in nano-biotechnology. In certain sponges these proteins are
responsible for silica biosynthesis. Experiments done in-vitro suggest that
the silicatein monomers self assemble in a fractal sheet intermediate ,
which due to certain thermal instabilities fold into long filaments. We are
interested to study this mechanism from the dynamics point of view to
explain the whole folding process. We have been able to model the process
and are working in order to address the global and local stability issues.
Nano particles concentration - Dr. Frederic Bottausci - Sophie Loire
Using the titanium device, we have experimentally shown the ability to concentrate Nanometer sized particles.
We use the combination of the flow motion generated by AC electric field and the DEP force. We have then
shown an increase of 25-30% in concentration of 10nm quantum dots and 15nm DNA in regions few microns in size.
