Center for Nonlinear Science

Center for Nonlinear Science
[1]
Center for Nonlinear Science
Contents
1  Vision
2  Major Research Components
A  Chaos in classical and quantum
systems
B  Dynamics of spatially extended
systems
C  Dynamics of biological systems
D  CNS Faculty
3  Education, Human Resources, Diversity, and Outreach
4  Shared Facilities
5  Collaboration with Other Sectors
6  International Collaboration
7  Seed Funding and Emerging Areas
References
1  Vision
Over the past two decades, investigations of nonlinear
systems have revealed that the simplest laws of nature can lead to bewilderingly
complex dynamics, and yet that such dynamics exhibit universal features which
are largely independent of the details of the underlying system. Thus, phenomena
as disparate as neuronal dynamics and pattern formation in fluids can be
studied with the same mathematical tools.
The impact of nonlinear science across the broad spectrum of natural, life,
and medical sciences is due to the unity of its fundamental concepts. Investigators
of nonlinear phenomena from diverse backgrounds are transforming important
problems once considered intractable, or even ill-posed, into promising new
fields. Methods developed in the study of nonlinear systems are now being
applied at the
Georgia Institute of Technology
(GT) and
Emory University
(EU) to such
areas as pattern formation, quantum theory, biomotor control, microfluidics,
dynamics of neuronal populations, biological fluids, and nonequilibrium transport.
These problems cut across many fields, and the progress in solving them requires
bringing together researchers from widely diverse fields, such as mechanical,
biomedical and electrical engineering, physics, applied and pure mathematics,
chemistry, biological and health sciences.
Since the strong nonlinear dynamics program at GT is located at one of
the country's leading engineering schools, in close proximity to Emory, the
home to one of the country's most eminent
medical schools
, we are in a unique
position to respond to the challenge posed by the complex systems found in
engineering and materials, life and medical sciences - recently identified
by the
National Research Council
] as one
of the six grand challenges for physics. We have established a cross-disciplinary
Center in Nonlinear Science
(CNS) which brings together faculty, researchers and students in basic sciences,
engineering, and medical sciences.
The greatest challenge of modern nonlinear physics is to develop theory
and applications of dynamics in systems with many degrees of freedom. We envision
a multi-pronged attack on this grand problem through three
cross-cutting
research efforts, with nonlinear science as the unifying theme
. The CNS
cross-disciplinary program will be based on common concepts unifying a broad
range of problems of basic science, such as pattern formation, nonequilibrium
dynamics, turbulence, classical and quantum chaos. Research and advanced
training fostered by CNS will impact on a diverse range of applications of
nonlinear science to fields such as biophysics, neuroscience, engineering
problems involving liquids, interface motion, novel materials, flows at microscales,
and nonlinear control, through the three research efforts:
1 -
Chaos: Classical and Quantum
Conceptually new methods, applicable to chaotic as
well as mixed chaotic/integrable high-dimensional systems, need to be developed.
Teams combining diverse talents will approach the problem from both the experimentally
driven (atomic physics, nonequilibrium dynamics, complex materials) investigations
of multi-dimensional problems, and by developing new mathematics (high dimensional
billiards, periodic orbit theory, wavelet methods) to analyze such problems.
The basic mechanisms of chaos in Hamiltonian and dissipative systems and
chaos-order transitions in multi-dimensional and infinite-dimensional systems
will be addressed both in quantum and classical contexts.
2 -
Dynamics of spatially extended systems
: Spatially-extended
systems are systems with potentially very many coupled degrees of freedom.
The dynamics of such systems can range from ordered (pattern formation near
onset) to very disordered (fully-developed turbulence). This research effort
seeks to break new intellectual ground by focusing on problems in three broad
fields at the interface between these two extremes, (a) pattern formation
in technological processes, (b) spatially-extended dynamics of living systems,
and (c) theory of spatiotemporal chaos. Cross-disciplinary interaction with
technology and biology has tremendous potential for achieving far-ranging
impact, both by stimulating development of novel methods of characterizing
spatiotemporal complexity, and by applying existing tools of nonlinear science
to important problems arising from experimental advances in these fields.
3 -
Dynamics of biological systems
: The rapidly evolving fields
of biophysics and bioengineering offer opportunities for advances over a
broad front. The proposed research effort will focus on investigations of
biological systems which lend themselves to dynamical modeling; ``many degrees
of freedom challenge'' in this context refers to modeling of small or large
arrays of excitable cells. In one direction, the research interactions will
be inspired by advances in physics: what are biological implications of ``stochastic
resonance'' and other surprising consequences of interplay of stochasticity
and nonlinear dynamics discovered by physicists? In the other direction,
rapid advances by CNS experimentalists engaged in neuromorphic engineering
of small neural networks in hardware, spatiotemporal imaging of human brain
activity, and construction of novel hybrid systems comprised of synthetic
neurons coupled to living tissue, will remain a continuous source of challenges
for nonlinear theorists.
The innovation we bring to this multi-pronged research effort is
training
in cross-cutting methodologies
. The key mission of CNS and a significant
fraction of its resources will be devoted to integration of diverse research
efforts through a common over-arching nonlinear science training, seminar,
workshop and visitor program. GT and EU have excellent faculty and research
infrastructure in nonlinear science, and the appropriate mix of specialties
to offer interdisciplinary research training. Nonlinear science training equips
students with a set of tools to formulate and solve diverse problems, wherever
they might arise: science, engineering, medicine. CNS faculty already has
considerable experience in initiating cross-disciplinary research projects.
The initial topics studied at the start of the CNS will of necessity reflect
the current interests of the individual faculty members, the CNS grant will
act as a vehicle for the generation of new interactions, and the development
of applications of nonlinear science to new areas. Co-advising of students
and research associates will foster training across diverse research groups,
alleviating the tendency of the traditional model to turn students into clones
of their advisors. CNS will stimulate cross-disciplinary research and communication
skills through intensive project-based courses, in which small teams investigate
topics guided by faculty members with complementary perspectives. Furthermore,
CNS will generate a highly cooperative and diverse research environment through
cross-departmental research seminars, student-run seminars, regional workshops,
yearly retreats, and an active visitor program.
Strategic impact and outreach
Why is CNS based at Georgia Tech?
GT is the leading science and technology institution in the Southeast, with
strong state and industrial support. It is a first-rate
engineering school
, in the process
of rapid growth, with CNS profiting from the ongoing expansion of the unique
GT/EU common
biomedical engineering department
the ongoing GT expansion in
nanosciences
and
biosciences
, GT's
supercomputing facilities, and close proximity of and institutional affiliation
with the
Oak Ridge National Laboratory
a leading institution of basic and applied research. The CNS graduate training
and research program will provide a unique common platform bringing together
faculty and students across disciplinary boundaries, will establish bridges
to other research institutions in the Southeast, will outreach teaching and
minority colleges and industrial sites, and stimulate progress in the field
nationally and internationally through a visitor program, exchanges and focused
workshops. The GT/EU's
Atlanta
location, with easy access to one of the world's busiest
airports
, makes it ideal for visitors
and workshops, and it already serves as a meeting place for researchers from
universities in the South, as well as a site for major international conferences.
The broad range of CNS activities, while highly desirable, cannot be sustained
by individual grants. The very existence of the proposed program hinges on
the availability of cross-disciplinary funding, something that only CNS can
make possible. The CNS program will be instrumental in fusing the faculty
from different schools at GT and EU, which will in turn facilitate research
training by allowing the CNS faculty to recruit, train, and financially support
postdoctoral associates and graduate students for the express purpose of pursuing
cross-disciplinary research projects in nonlinear systems. In the long run,
CNS will seed a basic institutional change, with faculty from diverse specialties
sharing post-docs and students and building collaborations that will far
outlast the CNS grant itself.
2  Major Research Components
We highlight here CNS' Major Research Components (MRCs) and the ways in
which they are interconnected, with special emphasis on connections that
would gain strength and coherence through CNS. Throughout the text, the contributing
faculty (see sect.
are indicated by initials; [LAB] stands for L.A. Bunimovich, and so on. The
faculty whose primary responsibily is managing a given MRC are indicated in
bold type in the ``senior personnel'' lists.
A  Chaos in classical and quantum systems
SENIOR PERSONNEL:
P. Cvitanovic'
J. Bellissard, L.A. Bunimovich
R.F. Fox,  R. Grigoriev,
R. Hernandez, M. Schatz, A. Shilnikov,
T. Uzer
and K. Wiesenfeld
OTHER PERSONNEL: Total of 7-9 research associates and 10-14 graduate
students at any given time.
Overview:
This MRC's strategy is two-pronged - combine physics and
chemistry experts in applications of quantum mechanics and nonequilibrium
statistical mechanics into teams with mathematicians and mathematical physicists
working on developing new methods. The goal is to address the same outstanding
grand challenge of nonlinear physics: describe classical and quantum chaotic
dynamics in dimensions higher than what has hitherto been possible.
Chaotic dynamics of systems with effectively few degrees of freedom is
today well understood, and forms the basis for current applications of nonlinear
methods to a broad range of fields. However, when the dynamics involves many
strongly coupled degrees of freedom, available tools for describing chaos
largely fail. Experiments such as the high-resolution spectroscopy of highly
excited (so-called ``Rydberg'') atoms in strong external fields pose both
a formidable challenge to theorists and at the same time offer ideal physical
laboratories for investigating multi-dimensional dynamics. Developing the
theory for these experiments has so far been stymied by the fact that the
classical dynamics undergoes a radical change when the number of degrees
of freedom exceeds two: beyond that threshold, a wealth of new physics becomes
possible. This inability to handle high-dimensional chaos in a systematic
way remains a fundamental barrier to the application of the methods of nonlinear
science in many disciplines.
Forays into this vast region are still comparatively rare, partly because
beyond this divide we lack diagnostic tools comparable in power to the Poincaré
surface of section method, and conceptually new methods, applicable to chaotic
as well as mixed chaotic/integrable systems, need to be developed. This MRC
seeks to break the impasse by putting together teams combining diverse talents
in order to approach the problem from two ends: (a) experimentally driven
(atomic physics, nonequilibrium dynamics, complex materials) investigations
of multidimensional problems, and (b) new mathematics (higher dimensional
billiards, periodic orbit theory, wavelet methods) to analyze such problems.
Here the cross-disciplinary interaction provided by the center has tremendous
potential for achieving far-ranging impact. Thus, one long-range goal of
this MRC is to use complex systems studied experimentally in physics and
chemistry to stimulate development of novel mathematical physics approaches
to describing and utilizing high-dimensional chaos. The second long range
goal of this MRC is to identify problems in other fields that could significantly
benefit from the application of advances in the theory of high-dimensional
dynamics.
The cross-disciplinary focus group on
Quantum chaos in high-dimensional
dynamics
brought together by CNS will attack a number of experimentally
measurable manifestations of quantum chaos where traditional approaches fail.
Specific examples include stochastic ionization of atoms in rotating microwave
fields, chaotic scattering of electrons, and wave chaos in elastodynamic media.
At present, the recently developed phase-space transition state theory of
transformations (
e.g.
, chemical reactions), the periodic orbit theory
and multi-wavelet methods show great potential, both as diagnostics of chaos
in higher dimensions, and as new ways to understand detailed high-dimensional
dynamics. Advances here are expected to impact a diverse range of problems,
including (to name but a few) the rearrangement reactions of atomic nanoclusters,
ballistic electron transport in mesoscopic systems, and the
capture of comets and asteroids.
The second focus group on
Chaos and chaos-order coexistence in high-dimensional
dynamics
will expand the basic mechanisms of chaos in Hamiltonian and
dissipative systems and chaos-order transitions in multi-dimensional and infinite-dimensional
systems along several directions. The goal of the first of these directions
is the construction of new classes of high-dimensional classical billiards
exhibiting chaotic behavior. The second closely related problem is the determination
of quantum properties in high-dimensional chaotic billiards. The third is
experimentally driven: CNS/EU experimental studies of dynamics of complex
materials (glasses, foams, colloidal suspensions, granular materials) and
the CNS numerical studies of chemical nonstationary stochastic dynamics require
the development of new methods for effective reduced-dimensionality descriptions
of ``hybrid'' dynamical models.
Why is the CNS necessary for the success of this MRC?
The proposed
CNS will provide the kind of support not available from individual grants,
but that is vitally important to initiate, stimulate and sustain cross-disciplinary
research. Without the CNS, both young faculty - afraid to gamble with their
tenure decision- and senior faculty -too entrenched in their fruitful but
narrow subdisciplines - are unlikely to entertain the radical or risky collaborative
projects that will likely be necessary to make advances in high-dimensional
dynamics. In particular, the CNS will serve as a meeting place for faculty,
students and post-docs interested in initiating new cross-disciplinary research
directions. CNS post-docs will be recruited not for particular faculty but
to solve collaborative problems within this MRC. The CNS senior faculty not
directly involved - but firmly committed to supporting such research initiatives-
will nonetheless provide mentoring at the crucial early stages of collaborative
projects. Moreover, the CNS will offer infrastructure to support the new
research directions on high-dimensional chaos through workshops, short-term
visitors and long-term visitors both from other nonlinear science research
centers, and from institutions which could potentially apply the new methods
to outstanding challenges in other disciplines.
Background.
GT has been at the forefront of classical and quantum
chaos research since the very inception of the field, pioneered by (among
others) GT researchers J. Ford, L.A. Bunimovich and P. Cvitanovic'. Instead
of using space here to detail the past research of the senior investigators,
we pick one recent delightful collaboration (in which, figuratively, Lagrange
meets Bohr in Outer Space) as an illustration of how a cross-disciplinary
team such as the teams proposed here can solve problems that no individual
researcher could have solved alone.
Launched on August 8, 2001, NASA's Genesis spacecraft is currently on a
mission to collect solar wind samples. M. Lo of the Jet Propulsion Lab, who
led the development of the Genesis mission design, has recruited Caltech mathematician
J. Marsden, GT physicist T. Uzer, and West Virginia U. chemist C. Jaffé,
to help in the trajectory design. Why? The Genesis trajectory is by design
a highly chaotic orbit of the infamous three body problem, controlled by
the Lagrange equilibrium points. The same dynamics controls the motions of
comets and asteroids: some of the most dangerous near-earth asteroids and
comets follow similar chaotic paths.
In what would have been a surprising turn of events for anyone but N. Bohr,
the identical type of chaotic orbits that govern the motions of comets, asteroids,
and spacecraft, are being traversed by highly excited Rydberg electrons on
the atomic scale. The near-perfect analogy between the atomic and celestial
mechanics implies the transport mechanism for both systems are virtually
identical: On the astronomical scale, transport takes the spacecraft from
one Lagrange point to another, until it reaches its desired destination. On
the atomic scale, transport refers to the electron initially being trapped
around the atom and then escaping by ionizing, never to return. The orbits
used in celestial mechanics to design space missions turn out to also determine
the ionization rates in atoms or rates of chemical reactions in the case
of molecules! Such connections between micro and macroworlds are not only
of great intellectual interest, but have important implications for engineering
in the aerospace and chemical industries. And the solution of this problem
is greatly enriched by the collaboration of this
ad hoc
group of chemists,
physicists, and mathematicians. CNS will ensure that such groups would be
formed frequently and that they would persist long enough to advance the field.
Focus group: Quantum chaos in high-dimensional dynamics
GT has been at the forefront of research into quantal manifestations of
classical chaos for over two decades. The research has been particularly fruitful
on ``Rydberg'' atoms in which an electron is promoted to such a high energy
state that it almost becomes a classical object. Rydberg atoms and molecules
represent an extreme form of matter: They can be as large as a fine grain
of sand, can outlive excited states of ordinary atoms by many orders of magnitude,
and at the same time they are extremely sensitive to certain perturbations.
They are ideal physical laboratories for investigating multi-dimensional
dynamics.
The CNS cross-disciplinary framework will enable us to put together a
team whose goal is to solve a number of open challenges in quantum chaos
posed by the recent and planned experiments: ionization of Rydberg states
in rotating microwave fields, the chaotic scattering of Rydberg electrons,
applications of novel wavelet methods to analysis of the dynamics of multi-dimensional
systems, and closely related ``wave chaos'' phenomena that arise in studies
of elastodynamic resonators [
]. The track record so far is that such
problems have been successfully solved only through collaborative efforts
of atomic physicists, chemists, nonlinear dynamicists and mathematicians.
This is scenario is currently employed by CNS: TU and plasma spectroscopist
E. Oks (Auburn U.) are combining their expertise in atomic physics with the
renormalization-theory expertise of the CNS Ford Fellow C. Chandre, and the
geometrical expertise of S. Wiggins (Bristol U.), to solve a number of problems
- detailed below - that each of the investigators alone would not be able
to solve.
Most early knowledge about quantum chaos can be traced back to two fundamental
experiments, both of which were performed on Rydberg atoms [
]: The first one concerns quasi-Landau oscillations
in Rydberg atoms placed in strong magnetic fields (the quadratic Zeeman effect,
also known as the diamagnetic Kepler problem) and the second is ionization
of Rydberg atoms in linearly polarized strong microwave fields. The interpretation
of the quasi-Landau oscillations in terms of a particular periodic orbit
of the Hamiltonian - an elementary example of Gutzwiller trace formula [
] - ushered applications of classical mechanics
to a wide variety problems, which until that time had been considered the
exclusive domain of quantum theory-a very fruitful approach from which we
are still benefitting. The interpretation of the microwave ionization problem
remained a puzzle to atomic theory until its stochastic, diffusional nature
was uncovered through the theory of chaos.
The classic experiments in quantum chaos have been understood through models
with two or fewer degrees of freedom. However, a number of recent experiments
have broken new ground in the dynamics of multi-dimensional chaotic systems.
Among these are the high-resolution spectroscopy of Rydberg atoms in crossed
static electric and magnetic fields [
], and the microwave ionization of hydrogen
using elliptically polarized radiation [
10
]. Developing
the theory for these experiments is an outstanding challenge of nonlinear
dynamics.
Chaotic ionization of electrons in microwave fields:
In recent experiments
probing multidimensional chaotic systems, a strong static electric field
is added parallel to a linearly polarized microwave field of comparable strength
11
]. These detailed experiments [
11
] show ionization yield curves which are rich with
regular oscillatory features and signatures of resonance transitions. Since
the fields are strong enough to render the standard time-dependent perturbation
theory useless, a new approach is needed to sort out the many transitions
that arise from the interplay of the strong static fields and the dressed
states created by the strong microwave field. The CNS group has recently
shown that most of the experimental observations can be satisfactorily understood
in terms of multi-frequency transitions driven by a single-frequency microwave
field between Floquet (or quasi-energy) states [
12
].
Unlike the linear and circular polarization cases, the elliptically polarized
case [
10
] supports no constants of motion, and explores all
three space dimensions - too many for the Poincaré surface of section
method to be of any use, and is therefore an ideal quantum chaos experiment
to which the time-frequency wavelet transform methods [
13
] can be applied. Indeed, there is still no satisfactory
classical mechanical treatment of this problem, and we intend to find the
sources and mechanism of chaos using the time-frequency wavelet transform,
to be explained below.
Chaotic scattering of electrons in three dimensions:
Quantal evidence
for chaotic scattering in the crossed-fields problem was soon confirmed by
classical calculations showing that the ionization times for the electron
showed fractal structure. By concentrating on the threshold ionization dynamics
14
], the CNS group was able to identify the classical
mechanism that explains this and related observations for energies below,
at, and above threshold. In particular, it was found that the transition
to chaotic scattering is caused by a critical point in the Hamiltonian flow
(the Stark saddle point), which in turn arises from a velocity-dependent,
Coriolis-like force in Hamilton's equations [
]. The CNS group
has shown that there exists a ``transition state'' in the planar atom in
crossed electric and magnetic fields problem [
14
] . The fundamental
importance of the transition state is that it represents a state of no return.
By means of this object, the coordinate space is partitioned into two regions:
one corresponds to atomic states, while the other corresponds to ionized
states.
Until very recently, neither the theoretical understanding nor the computing
power was adequate to explore phase-space transport beyond two degrees of
freedom. In two degrees of freedom, there is a distinguished periodic orbit
whose projection into coordinate space connects two relevant branches of
the equipotential surfaces. This unstable periodic orbit is called ``periodic
orbit dividing surface'' because it bounds a surface separating reactants
and products. Recently a CNS/Bristol team has generalized this solution by
constructing hypersurfaces of no return in the
phase space
of strongly
coupled, multi-dimensional systems [
15
]. The solution leads naturally to
the multi-dimensional generalization of a two-dimensional saddle point and
its associated separatrices. Indeed, the advances in dynamical systems theory
16
], in particular the concept of ``normally hyperbolic
invariant manifolds'' [
17
], anchor rigorously the notion
of a ``barrier'' in phase space of the classical theory of chemical reactions
in nonlinear dynamics.
The CNS group will apply these ideas to a wide variety of physical problems,
since transition state is a general feature of many dynamical systems, provided
that the system can evolve from ``reactants'' into ``products''. The transition
state method, therefore, is not confined to chemical reaction dynamics, but
it also controls rates in a multitude of interesting systems, including the
rearrangements of clusters, the ionization of atoms [
14
], conductance due to ballistic electron transport
through microjunctions, and the capture of comets and asteroids [
18
] (described above in the introduction to this MRC).
Wavelet methods:
Much of our understanding of nonlinear systems
is based on their Fourier spectra, and especially the resonances between different
modes of the system (KAM theorem, Fermi resonances in molecules, Laskar's
work in celestial mechanics). Frequency analysis of quasi-periodic systems
shows that the motion can be trapped in nearly quasi-periodic resonance zones.
These successes suggest that snapshots of a chaotic system in terms of frequencies
13
] could provide key information about dynamics also
in multi-dimensional settings, where the conventional methods of analysis
fail us. The recent work by the CNS group demonstrates that one can take
snapshots of a chaotic system in terms of its time-varying (instantaneous)
frequencies. One of the goals of this focus group will be to develop wavelet
methods which would allow us to zoom into segments of chaotic trajectories
with arbitrary frequency resolution.
Focus group: Chaos and chaos-order coexistence in high-dimensional dynamics
Chaos theory deals with systems with complex dynamics. Therefore it is
especially important to have a rich collection of physical systems which can
be fully (best of all, rigorously) investigated and thus provide a firm foundation
for intuition required to deal with complicated, ``real" systems. In Hamiltonian
dynamics
billiards
form such distinguished class of physical systems.
Besides the role they play in optics, acoustics, classical mechanics, statistical
mechanics, etc., billiards are a very useful testing ground for new approaches
in quantum chaos.
In Hamiltonian systems there are two basic mechanisms of chaos: dispersing
and defocusing. The first one has been discovered by Hadamard, Hedlund and
Hopf in geodesic flows on surfaces of constant negative curvature, and by
Sinai in billiards with dispersing boundaries. The billiards whose boundaries
are focusing (such as circles, ellipses, spheres, ellipsoids, surfaces of
positive curvature) were believed not to exhibit full chaos until Bunimovich
discovered in 1974 that Hamiltonian systems can reach chaos by another mechanism,
``defocusing'', whereas for sufficiently long free paths ``focused'' beams
start diverging. All of the original examples were two-dimensional and for
some 25 years a question whether the mechanism of defocusing can also wreck
chaos in higher dimensions remained wide open. The trouble lies in a well
known optics phenomenon, astigmatism, caused by wide variations in the strength
of focusing along different reflection planes. Recently the CNS group [
19
]-[
21
] has
proved that the mechanism of defocusing can generate chaos in any dimension
for a particular class of billiards. The CNS team will attack this problem
from two directions. The first one deals with classical chaos: the goal is
to extend the class of high-dimensional billiards with chaotic behavior, employing
the boundary focusing components of constant positive curvature. The guiding
conjecture is that focusing components of the boundary must be also absolutely
focusing.
The second natural problem is to study quantum analogs of high-dimensional
> 2) chaotic focusing billiards. Recently T. Papenbrock introduced
a provocative mechanical model of nuclei [
22
], a model
equivalent to a high-dimensional focusing billiard. Numerical experiments
show that these billiards are chaotic, and the quantum chaos investigations
23
] of these systems indicate very interesting behavior.
The CNS group will will study a much larger class of such multi-dimensional
systems and develop rigorous foundations for their quantum-chaotic properties,
properties so far only suggested by numerical investigations.
Generic Hamiltonian systems are neither integrable nor chaotic; their phase
space is a mixture of integrable islands and chaotic components. Even though
this behavior was proven to exist for a generic Hamiltonian systems and was
found in numerous computer experiments (the standard map is the best known
example), there are no rigorously investigated examples of physical systems
with such behavior. Recent developments [
24
] suggest that it should be possible to exploit
such
mixed phase-space
properties to design systems for which the energy
of a source (such as a beam of light) is arbitrarily nonuniformly distributed
across the configuration space, opening up a wealth of applications, such
as in the stochastic cooling of atoms [
25
].
Either purely deterministic dynamical systems or stochastic random process
are traditionally employed to model physical, chemical and biological phenomena
and processes. A theory of either is very rich, and our intuition about evolution
of such systems is well developed. This intuition is based on explicitly
solvable simple examples and physically transparent (but nontrivial) systems.
In the theory of stochastic processes the examples are sequences of identical
independently distributed random variables (Bernoulli shifts), random walks,
etc. In dynamical systems theory such explicit models include toral automorphisms,
one-dimensional quadratic maps, billiards, etc.
Many physical problems are
hybrid systems
in the sense that their
dynamics is neither purely deterministic nor purely random but rather a combinations
of both. This holds for disordered systems, chemical kinetics, and theoretical
problems of computer science (Turing machines with many heads and/or many
tapes). Complex materials such as glasses, foams, colloidal suspensions,
and granular materials with liquid-like structures and solid-like properties
studied experimentally at microscopic level by the CNS/EU group by means
of confocal microscopy in both static and dynamic settings [
26
27
], offer a rich set of challenges for ``hybrid''
systems theory. Further inspiration can be garnered from the CNS group numerical
studies of
nonstationary stochastic dynamics
. In some cases, the dynamics
of reduced-dimension coordinates describing an effective solute may induce
nonlinear responses in the effective solvent. In
spatially heterogeneous
solvents
, models have been developed using the generalized Langevin equations
with space-dependent friction. When the solute is sufficiently concentrated,
the collective solute dynamics can be modeled by a time-dependent self-consistent
friction [
28
29
].
The time evolution of such systems is often quite counter-intuitive [
31
]. Traditionally one considers either a small random
perturbation of a deterministic system, or by adding a small advection component
to a diffusion process. Such small perturbations do not address the hard
problem, the behavior of hybrid systems whose evolution is governed by deterministic
and stochastic components of comparable strength. There exist very few (if
any) comprehensive investigations of such hybrid systems. Recently a breakthrough
has been achieved in the studies of one class of such systems, known either
as the ``Lorentz lattice gas cellular automata'' or the ``Many-dimensional
Turing machines''. Let a light (particle, signal, wave, spike, animal, etc.)
move on some graph. there is a scatterer (scattering rule, protocol etc.)
which determines where this propagating object will go after visiting a given
vertex. If the scatterers are randomly distributed, the dynamics is a deterministic
walk in a random environment. The moving object can in turn affect the environment
by changing the state of a visited vertex. Then one can study not only motion
of a single particle (like in the classical Lorentz gas) but also dynamics
of many moving objects which interact indirectly, by changing the environment
in which another objects also move. This challenging high-dimensional systems
is also a natural high-dimensional representation of a Turing machine because
each vertex can be thought of as an infinite tape with a (deterministic)
protocol which prescribes how environment should change in a given vertex.
This model happens to be the central concept of the new branch of theoretical
computer science called ``collision based computing'' [
30
]. Deterministic walks
in random environments also have applications in statistical physics, chemical
kinetics, environmental studies, neuronal dynamics, etc.
Consider now walks in ``rigid'' environments [
31
] where
one introduces a parameter
of environmental's rigidity into a general
deterministic walk in a random environment, with the environment of a vertex
changing after
th visit to this vertex. It turns out that such walks
are completely solvable models on one-dimensional lattices. Moreover, they
exhibit the three possible types of diffusive behavior: normal, sub- and
super-diffusion. All these types of behavior are deterministically generated,
demonstrating the interplay between symmetry properties of a lattice and
the types of scatterers which form the environment [
32
]. Another unexpected phenomenon exhibited by random
lattices is localization [
33
], analogous to the Anderson's localization in condensed
matter physics. This research will continue in several directions: studies
of walks in environments with nonconstant rigidity, studies of models with
many particles and continuous limits of these models.
While in the context of quantum theory Hamiltonian systems play a special
role, dissipative systems are at least as important as models of fluid, chemical
and neuronal dynamics. It is increasingly clear that it is quite unlikely
that hyperbolic attractors appear in models which have anything to do with
real systems, and CNS group will therefore focus on pseudo-hyperbolic attractors
in high-dimensional systems, where ``pseudo" stands for chaotic attractors
with homoclinic tangencies but without stable periodic orbits. Another open
multi-dimensional challenge in dissipative dynamics is the description of
``hyperchaos'', i.e. chaos with more than one unstable directions. This situation
is relevant for a large variety of applications. Another direction for the
CNS teams is study of bifurcations in slow-fast systems which appear in neural
(individual and group) dynamics. The CNS would team the Georgia State University
mathematician with the EU experimentalists in order to investigate a fourteen-dimensional
neuron model [
34
].
B  Dynamics of spatially extended systems
SENIOR PERSONNEL:
M. Schatz
R. Grigoriev
G.P. Neitzel
, L.A. Bunimovich, P. Cvitanovic', K. Wiesenfeld
OTHER PERSONNEL: Total of 4-6 research associates and 8-11 graduate
students at any given time.
Overview:
Spatially extended systems are systems with potentially very many coupled
degrees of freedom. The dynamics of these systems can range from ordered (pattern
formation near onset) to very disordered (fully-developed turbulence). While
the methods of nonlinear science have achieved a certain ``universal'' (in
the spirit of equilibrium critical phenomena) understanding of low-dimensional
behavior in spatially extended systems, current tools for describing these
systems fail when the dynamics are high-dimensional, involving many spatial
and temporal degrees of freedom. This inability to handle spatiotemporal complexity
presents a fundamental barrier to the application of the methods of nonlinear
science in many disciplines
This MRC seeks to break new intellectual ground by concentrating efforts
in three focus groups: (a) Pattern Formation in Technological Processes, (b)
Spatially Extended Dynamics in Living Systems, and (c) Theory of Spatiotemporal
Chaos. Cross-disciplinary interaction in technology and biology has tremendous
potential for achieving far-ranging impact. Many important problems in technology
and biology involve dynamics that are typically in spatiotemporally complex
regimes. Thus, one long range goal of this MRC is to use studies of problems
in these fields to stimulate development of novel methods of characterizing
and classifying spatiotemporal complexity. At the same time, the methods
and approaches of nonlinear science are still relatively unknown in many
areas of technology and biology. Thus, a second long range goal of this MRC
is to applying existing tools of nonlinear science to important problems
arising from advances in technology and biology.
The recent establishment of the GT/EU Center for Nonlinear Science has
already helped initiate the kind of cross-disciplinary research envisioned
under this MRC, as described in detail below. Patterns in technology are being
explored by means of optical methods of multipoint manipulation developed
at the CNS for hydrodynamics experiments; these methods are providing the
means for conducting a wide range of novel experimental studies that include
exploration of dynamics and control of coating flow (a fundamental manufacturing
process) and application of pattern formation methodologies to microfluidics,
the science and technology of microscale fluid mechanics for biological and
chemical applications. Spatially extended dynamics of arrythmias in cardiac
tissue in vivo is a compelling problem that pushes development of new experimental
and theoretical approaches for characterizing spatiotemporal complexity.
The planned CNS provides the kind of support, not available from individual
grants, that is vitally important to stimulate, sustain and broaden such cross-disciplinary
research on spatially extended systems. Firstly, the CNS provides support
for the critical mass of personnel (students and postdocs) needed for work
on new cross-disciplinary research directions at the crucial preliminary
stages when such work lacks the track record needed to attract support from
more conventional sources. Secondly, the CNS will support the cross-pollination
of ideas for new research directions on spatially extended systems in technology
and biology by using short-term (e.g., workshops) and long-term (e.g., resident
scholars) visiting opportunities to attract to the CNS researchers from universities,
national laboratories, and industrial research centers. These visitors are
expected to benefit from CNS interaction by acquiring new approaches to unsolved
problems in their disciplines. At the same time, the research environment
at the CNS will be invigorated by visiting researchers who provide fresh
problems that push the current frontiers of our understanding of spatially
extended dynamics.
Focus group: Pattern Formation in Technological Processes
Dynamics and Control of Coating Flow:
Coating is a technological
process of great significance, with applications ranging from electronics
and optics to automotive and aerospace industry. A typical coating application
involves a two step process: first the substrate is coated with a layer of
solution, then the solvent is left to evaporate, leaving a layer of solute
on the surface of the substrate. The main difference between various coating
techniques is in the way the initial liquid coating is produced. For instance,
dip-coating technique is used for optical fiber coating [
39
],
and (anti)reflective optical coatings of lenses and mirrors. Similar techniques
are used to produce hydrophobic [
40
] and hydrophilic coatings. Spray
coating is used to produce sol-gel coatings of TV screens [
41
and, more routinely, for painting. Spin coating [
42
] which
was originally developed for microelectronics applications has also found
numerous applications in the optical industry.
Most types of coating techniques (e.g., dip-, spin-, or blow off coating)
involve forced spreading of the liquid onto a substrate, where the external
forcing is provided by gravity, inertia, viscous drag, or imposed gradient
in surface tension. Regardless of the type of forcing, the process of driven
spreading shows a generic feature: for some parameters the solid-liquid-gas
contact line exhibits a transverse (fingering) instability [
43
]. The instability is believed to arise due to
the increased mobility of the capillary ridge forming near the contact line
44
] and manifests itself in the formation of fingers
and troughs advancing with (in general) different velocities.
Another type of instability arises when the liquid coating starts to evaporate
r2.4in
Figure 1: Thickness profile of an evaporating liquid layer
on the surface of a disk spinning at 2000 RPM in the center (top) and 2cm
away (bottom) (Courtesy D. P. Birnie, U. Arizona).
45
].
Surface tension gradients arising from variations in temperature (thermocapillarity)
or concentration (solutocapillarity) can combine with the normal pressure
of the escaping vapor on the liquid-gas interface to destabilize the liquid
layers of uniform thickness. The ultimate fate of this latter instability
is eventually determined by the properties of the substrate: if it is non-wetting,
the film will generally rupture [
49
], producing a pattern of dry spots,
a phenomenon often referred to as de-wetting or reticulation. Otherwise it
may remain continuous but spatially nonuniform. For instance, during spin-coating,
instability manifests itself in a pattern of radially oriented lines of thickness
variation, or striations [
48
] (see Figure
).
Both fingering and evaporative instabilities can crucially affect the quality
of produced coatings, so understanding the governing mechanisms and devising
methods to suppress them is fundamentally and practically important. A number
of CNS faculty are actively involved in related research. The studies of
thermal convection in thin layers [
49
50
51
52
] and liquid bridges [
53
and control of interfacial instabilities [
45
54
] represent only a few of the examples of the previous
and current work. On-going research on coating flows is focused in two main
directions: dynamics and control of driven contact lines and thermo- and
soluto-capillary instabilities of thin liquid layers.
Perhaps the most familiar example of fingering instability is provided
by the formation of drip rivulets that can occur when vertical surfaces, such
as household walls, are painted. Although the case of gravity-driven spreading
is relatively well studied, the problem of surface-tension-gradient-driven
spreading has received comparatively little attention [
46
]. This latter case is important because experimental
approaches developed for manipulating thermocapillary convection (described
below) could find use in technological applications, so a much deeper understanding
of the fundamental physics of the contact line instability is necessary to
both exploit and suppress it. Our theoretical analysis [
47
] suggests that the presence of capillary ridge
determines the stability in this case as well. Experimental study of the
fingering instability will exploit the unprecedented capabilities of multipoint
thermal actuation to manipulate the dynamics of the coating flow.
The effectiveness of thermocapillary-based optical manipulation of dynamics
has been demonstrated in several previous experiments on free surface flows.
The imposed thermal profile was used as a tool for probing and controlling
instabilities leading to pattern formation. In work on surface-tension-driven
Bénard convection, selected patterns were imposed as initial conditions
using a computer-controlled optical heating from an infrared laser. In one
case, imposing patterns with designed imperfections permitted, for the first
time, quantitative measurement of the dynamics of penta-hepta defects, which
are the most common disordering mechanism in hexagonal patterns found in
a wide variety of physical systems [
52
]. In a
second case, effective closed-loop control was achieved for the suppression
of thermocapillary convection waves by using infrared temperature measurements
coupled with feedback via optical heating by an infrared laser [
55
].
r2.4in
Figure 2: Optically controlled spreading of a thin liquid film
on a solid substrate. The formation of ``fingers'' (right side of figure)
can lead to nonuniform coating. Small temperature-induced surface-tension
(thermocapillary) gradients, applied optically (left side of figure), can
suppress the instability, permitting the spreading front to remain flat and
to spread uniformly.
Our preliminary experiments show that variations in the thermal profile
imposed in response to the contact line distortion can be used to actively
suppress the instability in an all-optical setup (see Figure
). These experiments also raise a number of questions
regarding the optimal choice of the spatial structure of the applied perturbations.
These questions can only be answered by taking into account the full dimensionality
of the physical space (local changes in the temperature of the liquid film
produce fluxes in the direction of the spreading as well as transversely to
it) as well as the non-normality of the evolution operator (the standard linear
stability analysis provides a poor description of the dynamics). To find
the answers more efficiently the theoretical analysis will be tightly integrated
with the concurrent experiments. These studies will also address the question
of whether the proposed techniques can be used for controlled patterning
of the coating flow.
Although the problem of controlling evaporative instabilities in pure liquids
is now rather well understood [
45
], essentially all liquids used in coating
applications are mixtures whose properties sometimes differ rather dramatically
from those of pure liquids. The studies involving complex (multi-component)
liquids have barely scratched the surface. The major difficulty in describing
the complex liquids is the fact that all physical properties become functions
of relative concentrations of different components, immediately increasing
the dimensionality of the problem. Additional complication is produced by
the presence of components that tend to segregate near the interface (e.g.,
surfactants). In order for the results of experimental and theoretical studies
to advance the state of knowledge in practically relevant cases, these difficulties
have to be addressed.
Combined theoretic and experimental approach will also be used for the
study of the dynamics and control of thin layers of multiple-component volatile
liquids. The theoretical description will be built by generalizing the approach
developed for pure liquids by RG and it will be extensively tested experimentally
in the laboratory of MS with the experiment and theory providing continuous
feedback for each other. The experiments will start by providing experimental
validation for the theory [
45
] developed for the case of pure liquids
and then move on to the more complicated case of binary mixtures and solutions.
Both RG and MS will interact extensively with the materials science group
at the U. of Arizona (D. P. Birnie, D. E. Haas), which has expertise in the
studies of spin-coating involving complex liquids.
Microfluidics for Biological and Chemical Applications:
Miniaturization
in electronics has spurred many advances in solid state physics and numerous
wonders of technology. Similarly, the goal of miniaturizing biological and
chemical processing (``Labs-on-a-Chip'') is stimulating both science and
technology in the field of microfluidics. To perform genetic assays, chemical
synthesis or water quality tests on a chip-sized device requires the ability
to manipulate a large number of tiny fluid samples. At present, there is
no broad agreement on the best methods for controlling fluid flow at the
microscale [
56
]. Techniques drawn from
studies of spatially extended systems, which, at first glance, are seemingly
unrelated to microfluidics, have, in fact, the potential for providing new
approaches for this emerging area.
r4.5in
Figure 3: Optically controlled thermocapillary driving of microstreams
on a FEATURELESS horizontal glass substrate is shown in a sequence of time
lapse images. A spatially varying temperature is imposed optically on the
substrate, inducing surface-tension gradients that (a) draw two 500
m wide silicone oil microstreams from reservoirs,
(b) turn the microstreams toward one another, (c) merge the microstreams,
and (d) drive the merged stream.
Lack of reprogrammability/reconfigurability is a major shortcoming of most
current microfluidic devices. Microfluidic chips today are constructed with
lithographic techniques inspired by microelectronics. However, unlike semiconductor
manufacture, where the resulting devices (e.g., CPUs) can be readily reconfigured
for many tasks, lithography in microfluidics etches the devices into a fixed
configuration that is typically capable of performing only a single very
specific assay. (If microcomputers had developed this way, a new chip would
have to be plugged in with every software installation!)
The possibility of using dynamics rather than geometry for microfluidics
is suggested by the work on coating flow described earlier. Preliminary experiments
demonstrate that optically-induced thermocapillarity can confine and transport
microflow on a substrate that is mechanically and chemically featureless
(Fig.
). In other words, no etching of pipes or
hydrophillic/hydrophobic surface treatments are required to channel the flow.
Instead, the fluid is confined and guided by a combination of ``self-containment''
by capillary forces along with dynamically adjustable thermocapillary forces
imposed by the illumination. Liquids in microdroplet form may also be driven
optically on liquid substrates. In Figure
, the
focused beam of an infrared laser is rastered along the liquid substrate
near the droplet and the beam heats the interface by direct absorption. The
resulting thermocapillary flow produces an interfacial motion away from the
laser spot that carries along the droplet at speeds of up to a few centimeters
per second. Again, no pipes or patterning are required to contain the flow;
the microdroplet is self-confined by its own surface tension.
Tools developed from studies of other spatially-extended systems can be
used to address important issues in dynamics-based microfluidics. For example,
studies of pattern formation in the classic Bénard thermal convection
problem have shown that thermocapillary-induced instability can lead to topological
changes,
e.g.
, break-up and rupture of the liquid [
49
]. These effects are well-modeled by both linear
and nonlinear theory in the convection problem [
57
]. Similar mechanisms
are expected in thermocapillary-driven microfluidics and can potentially
be used to split microstreams or microdroplets for metering purposes. As
a second example, studies of spatially-extended fluid systems have demonstrated
the phenomenon of chaotic advection, in which a passive scalar (e.g. concentration)
in the flow can exhibit complex behavior even in cases when the velocity
field of the flow
r4.2in
Figure 4: Optically controlled thermocapillary guiding and merging
of 300 nanoliter insulin droplets suspended on a perfluorocarbon liquid substrate.
A single droplet is driven in a repeated elliptical trajectory, as shown
in the multiple exposure image in (a). A time sequence of images shows the
merger of two droplets (b).
is not chaotic [
58
59
60
].
In many cases, chaotic advection can be described using methods developed
for understanding chaos in Hamiltonian systems [
58
]. Implementing chaotic
advection in microfluidics should beneficially enhance mixing, which otherwise
is usually dominated by slow molecular diffusion. Chaotic advection has been
implemented in geometry-based microfluidics [
61
] and can be implemented
in dynamics-based microfluidics by suitable temporal/spatial modulation of
the thermocapillary driving of the microflow.
A recently formed partnership between GT/EU CNS (involving MS, RG, GPN)
and Yerkes Regional Primate Research Center/Emory School of Medicine has proposed
to explore the potential of dynamics-based microfluidics for use in biomedical
applications. This partnership, which joins expertise in pattern formation,
hydrodynamics, and genetics, will explore the scientific and technological
issues related to developing a reprogrammable/reconfigurable PCR (Polymerase
Chain Reaction) analyzer for the identification and detection of infectious
agents.
Focus group: Spatially Extended Dynamics in Living Systems
Sudden cardiac death is the leading cause of death in the industrialized
world with the majority of such tragedies due to ventricular fibrillation
(VF). VF is a frenzied and irregular heart rhythm disturbance that quickly
renders the heart incapable of pumping blood and hence sustaining life. Instead
of contracting regularly and uniformly, the ventricles writhe and fibrillate
at a frequency some ten times faster than the normal heart rate. Understanding
these self-sustaining dynamics and the mechanisms responsible for their initiation
is crucial for developing effective and reliable defibrillation techniques.
The presently existing techniques are based on ``resetting'' the heart with
a strong electrical discharge and have large failure rates.
Although at first sight the dynamics of the heart tissue during VF seem
to be very irregular, it is not random and possesses a high degree of spatial
and temporal coherence, indicating that it is governed by a deterministic
process [
62
]. The dominant approach toward characterizing this
dynamics has been based on the studies of simplified models of excitable
media that all share with the heart the functional properties of excitability
and refractoriness. The generic feature of such models is that their dynamics
are dominated by travelling waves. In particular, reentrant spiral waves,
seen in numerical solutions of three-dimensional simplified models of cardiac
tissue for a long time were believed to occur during ventricular tachycardias.
However, experimental detection of such reentrant waves in fibrillating mammalian
ventricles has been difficult.
Recent experiments conducted by WLD and collaborators [
63
] suggest that spiral waves are rarely observed.
In the early stages of VF, the dynamics is dominated by transiently erupting
rotors (source structures surrounding the core of rotating spiral waves).
This activity is characterized by a relatively high spatiotemporal cross
correlation. During this early fibrillatory interval frequent wavefront collisions
and wavebreak generation are also dominant features. In the contrast, the
later stages (chronic VF) are characterized by patterns which are much more
complex and are less correlated in both time and space, while the epicardial
rotors are no longer observed.
Despite this complexity, some spatial correlation remains, indicating the
presence of spatially coherent structures (CS). Identification of such CS
was suggested as a general way to reduce the dimensionality of spatiotemporally
complex dynamics. This approach which was originally developed in the context
of fluid turbulence [
64
] has found numerous applications
in other areas, most notably characterization and control of chemical reactions
66
67
]. During the early stages of VF in the heart
tissue CS can be associated with rotors. Even though the spiral wave may
break down just outside the core, the rotors can be easily identified as
the points of phase singularity, or topological defects around which the
excitation wave propagates in clockwise or counterclockwise direction. High
degree of spatial and temporal coherence can be exploited to design implantable
defibrillators, which are capable of quickly detecting VF.
The origin of coherent structures present at the later stages is presently
unknown, which makes defibrillation during chronic VF especially challenging.
However, this is precisely the regime that should be targeted by clinical
defibrillation techniques, since by the time VF is detected in most patients
the earlier stages will be over. The lower degree of coherence indicates that
the dynamics is characterized by a larger number of degrees of freedom, making
it harder to analyze and control.
An additional difficulty is associated with the fact that presently no
general techniques exist capable of suppressing spatiotemporally chaotic behavior
in systems of complexity comparable to that of the heart tissue. One promising
approach is currently under investigation by RG and MS (from theoretical
and experimental standpoints, respectively). The main idea is to simplify
the description of the dynamics by factoring out the degeneracies associated
with translational degrees of freedom and performing statistical analysis
on the reduced data set using the proper orthogonal decomposition (POD) to
identify the coherent structures (Fig.
), their position,
and orientation, which will allow the construction of a reduced-order model.
Such a model can be used for the purpose of designing a control algorithm
for defibrillation in both the early and later stages of VF.
r2.2in
Figure 5: The fastest growing mode of the instability leading
to roll breakup in Rayleigh-Bénard convection.
Presently this approach is developed using Rayleigh-Bénard convection
(RBC) as a model system displaying spiral defect chaos (SDC). RBC represents
the typical features of spatiotemporally complex dynamics in general, such
as chaos in both the temporal and spatial domains, and interaction of coherent
structures at different length scales. The regime of SDC has particularly
many features similar to the dynamics of the heart tissue. The main attraction
of SDC is, however, that it is considerably more tractable than the fibrillatory
dynamics: its effective dimensionality is much lower, while the availability
of a quantitative weakly nonlinear description in terms of amplitude equations
provides the opportunity to model certain aspects of SDC analytically. In
addition, the dynamics of this system can be easily controlled on a very
fine scale via optically-imposed multipoint thermal actuation (see discussion
above).
The typical application of POD [
66
] uses time averaging in the absence of
external stimuli to extract coherent structures containing the effective
degrees of freedom from the chaotic signal. This approach is inherently flawed
for two reasons. First of all, (approximate) translational invariance produces
degeneracy in the position of coherent structures, smearing them over the
whole domain [
65
], rendering the dimension of the
embedding space produced by POD too large to be useful. Second, any external
stimulus (a requisite ingredient in any control, or in this case defibrillation,
technique) will knock the system out of its attractor, rendering the description
of the unperturbed dynamics largely irrelevant. Both of these problems can
be solved by changing the way in which statistical sampling is performed.
By replacing the time average with an ensemble average performed for a collection
of different initial conditions produced by imposing local perturbations,
a set of localized coherent structures is obtained, yielding a reduced-order
model whose dimensionality is independent of the system size.
The theoretical and experimental studies envisioned within this proposal
will look at implementing the above approach in the context of VF. RG will
be responsible for modeling and theory. The methodology will be tested experimentally
on convective systems in the laboratory of MS at GT and the results and techniques
will be adapted for application to the dynamics of heart tissue under the
direction of WLD, with experiments conducted at the clinical facility of
the University of Alberta. The experimental studies will concentrate on two
main directions: reconstruction and visualization of fibrillatory dynamics
in three dimensions, and characterization of the effects of spatially and
temporally localized external stimuli on the dynamics.
Since the ventricles are relatively thick, the influence of the third dimension
cannot be neglected, if an accurate description of the dynamics is to be
constructed. Since the proposed approach assumes that the data used to extract
coherent structures faithfully (and uniquely) represents the state of the
system, the availability of three-dimensional imaging with adequate spatial
and temporal resolution is crucially important. Equally important, reconstruction
and visualization of the dynamics of heart tissue in three dimensions is
necessary for gaining additional insights into the mechanisms leading to
the initiation and evolution of VF and for construction of more accurate
mathematical models. For instance, since the heart tissue is almost transparent
to intense radiation of appropriate frequency, optical tomography using voltage-sensitive
dyes can be employed for three-dimensional reconstruction.
The study of the response of the heart tissue to external stimuli represents
another important ingredient of the proposed approach. The inherent degeneracy
in the position (and possibly orientation) of coherent structures coupled
with their localized spatial structure, demands the ability to apply localized
perturbations to suppress the instabilities leading to VF. Novel ways to apply
temporally and spatially localized perturbations will be developed based
on the idea of optically activated electrical, thermal, or chemical stimuli.
Different methods will be tested experimentally and the best technique will
be identified and compared with the conventional technique of delivering electrical
stimuli via implantable electrodes. The analysis of the response to such
perturbations will be used to construct a reduced order model for designing
new defibrillation methods.
CNS would enable the integration of presently separate individual efforts
by MS, RG, WLD, and the clinical group at the University of Alberta (K. Kavanagh,
P. Penkoske) to study the dynamics of VF and enable the transfer of the expertise
developed in controlling spatiotemporally complex dynamics in convective
systems to the problem of designing reliable and efficient defibrillation
methods. This is an extremely complicated and ambitious goal, which cannot
be achieved on the level of individual collaborations. The success requires
a collaborative effort involving specialists with the expertise in the areas
of physiology, cardiology, spatiotemporal dynamics, and control. Without
such a highly centralized approach the progress will be limited to incremental
advances having little bearing on the ultimate goal - saving lives.
Focus group: Theory of spatio-temporal chaos
The GT experimental group of M. Schatz has a unique skill; they are able
to
design
a large repertoire of initial spatio-temporal patterns by
their multipoint thermal actuation technique (described above). If the dynamics
is spatio-temporally chaotic, such patterns are unstable and quickly fall
apart - and this instability is precisely what fascinates the mathematical
physicists which CNS will team up with this experimental group.
In explorations of modern field theories (classical Yang-Mills, gravity,
hydrodynamics, Ginzburg-Landau systems) the dynamics tends to be neglected,
and understandably so, because the wealth of their solutions can be truly
bewildering - the strongly nonlinear classical field theories are turbulent,
after all. If one is to develop a theory of a spatially extended systems that
are chaotic, one needs to determine, classify, and order by relative importance
the solutions of nonlinear field theories. Such systematic exploration has
so far been implemented [
68
69
] only for one of the very simplest field
theories, the 1-
Kuramoto-Sivashinsky system [
70
]. This research is still in its infancy, but it
has lead to a working hypothesis the CNS plans to explore: For any finite
spatial resolution, the system follows approximately for a finite time a
pattern belonging to a
finite
alphabet of admissible patterns, and
the long term dynamics can be thought of as a walk through the space of such
unstable patterns. The
periodic orbit theory
35
] provides sophisticated
mathematical machinery that converts this intuitive picture into a precise
calculation scheme. The patterns singled out by the theory can be created
and tested by MS's technique. The CNS team is currently the only one which
possess both experimental and theory expertise to carry this program to fruition.
Computer simulations of classical field theories are of necessity discretized
in space as well in time. They inspired research into Lattice Dynamical Systems
(LDS), systems of spatially interacting local units, with each unit in itself
a dynamical system,
e.g.
a nonlinear oscillator. So far, studies of
LDS have lead to a mathematically rigorous definition of space-time chaos,
and proofs of its existence for lattices of weakly interacting local chaotic
systems [
71
]. Another important advance
was the discovery that that the appearance of coherent structures can be
understood as a ``thermodynamic formalism'' phase transition [
72
], and existence of such phase transitions was
proven for weakly interacting LDS [
73
]. It is an experimental and numerical fact
that LDS and ``reaction-diffusion'' systems exhibit complex spatial patterns.
The traditional perturbation theory cannot be applied to these important
phenomena. The great challenge here is to develop a mathematically well-founded
theory, not only for its intellectual merits, but also because the problems
faced here might be well beyond what can be simulated by the conventional
numerical methods.
C  Dynamics of biological systems
SENIOR PERSONNEL:
R.L. Calabrese
K. Wiesenfeld
, G.S. Berns,
R.J. Butera, Jr., S.M. Potter, R.F. Fox, L.A. Bunimovich, P. Cvitanovic',
M. Schatz, R. Grigoriev
OTHER PERSONNEL: 1-2 research associates, 3-5 graduate students funded
by CNS each year: 3-4 research associates, 6-8 graduate students funded by
other grants
Overview:
The rapidly evolving fields of biophysics and bioengineering
offer excellent opportunities for research advances over a broad front. At
the same time, the interdisciplinary nature of the Center provides an unusually
attractive environment to tackle selected problems in these fields. CNS seeks
to capitalize on and strengthen existing ties between GT and EU in these
fields. Our combined effort in this area brings together researchers from
several GT units (biology, physics, and mathematics), the EU School of Medicine,
and the recently established joint GT/EU School of Biomedical Engineering.
This unit was formed in 1997 as a unique academic entity that evolved out
of a need to formalize collaborative research efforts and provide an innovative
educational forum between GT and EU. This academic unit reports to both the
College of Engineering at GT and the School of Medicine at EU. The School
of Biomedical Engineering takes advantage of the strong traditions of each
institution, enabling the School to define new degree programs that integrate
a rigorous grounding in life sciences and engineering, with real world experience
in industry and clinical medicine.
The proposed MRC will focus specifically on investigations of biological
systems which lend themselves to dynamical modelling. In this respect the
tools we will use include both traditional ones applied to new problems, and
newer tools that are themselves under continual modification and development.
In the former category sits modelling via nonlinear stochastic differential
equations and nonlinear time series analysis of very large, complex spatiotemporal
data sets, while the latter includes modelling of neuronal systems in VLSI
hardware, as well as hybrid systems which are part living tissue and part
electronic circuitry.
The emphasis on dynamical modelling nevertheless allows us to consider
problems that range in size and scope from subcellular processes to single
cell, interneuronal, and whole organ levels. The phenomena share a common
thread, complex, often chaotic and/or stochastic dynamics. Each of the projects
described in the following have in place both experimental and theoretical
components. The environment provided by CNS and its shared activities is
expected to greatly enhance the number and quality of these interactions.
The planned CNS provides the kind of support, not available from individual
grants, that is vitally important to sustain the inherently cross-disciplinary
research on biological systems envisioned here. CNS will provide a stimulating
environment for new research directions in biophysics and bioengineering by
supporting cross-disciplinary post-docs, students, short-term (e.g., workshops)
and long-term (e.g., resident scholars) visiting opportunities to attract
to the CNS researchers from universities, national laboratories, and industrial
research centers. These visitors are expected to benefit CNS interaction
by acquiring new approaches to unsolved problems in their disciplines. At
the same time, visiting scholars will invigorate the CNS research environment
by providing fresh problems and sometimes fresh approaches that push back
the current frontiers of dynamics in biophysics and bioengineering.
The research under this MRC falls into two focus groups, described in more
detail below. The first addresses problems of spatiotemporal dynamics in
neurobiological systems. The spatial aspect arises through interactions of
individual parts, in this case neurons. The fundamental scientific questions
and the tools we use to adress them depend on whether the number of interacting
elements is small or large. When only a few elements are involved, the specific
dynamical character of individual parts is important, and one seeks rather
complete information about the effects of parameter variation and feedback
control. It is here that analog VLSI and hybrid circuit methodologies are
especially powerful. When the number of interacting elements is large, in
our case ranging from several thousands (in neural tissue samples) to trillions
(whole brain studies), the central problem is different as one looks for
large scale emergent behavior. Here, it is imperative to develop new methods
for identifying structure within huge sets of spatiotemporal data. The expertise
of our research teams is well suited to tackling these challenges.
Focus group: Spatiotemporal dynamics of neurobiological systems
The projects within this focus group take advantage of the truly unique
composition of the CNS team, which includes researchers from engineering,
neuroscience, physics, and mathematics. Theoretical methods from nonlinear
systems mix with technologies of analog VLSI electronics, high speed data
acquisition and processing, fast optical imaging, and functional magnetic
resonance imaging. For this focus group, the epicenter for these efforts is
the Coulter Department of Biomedical Engineering, a joint venture between
the GT College of Engineering and the EU School of Medicine.
A primary focus of this department and the associated Laboratory for Neuroengineering
is the study of neurobiological systems combining the tools and techniques
of engineering and the quantitative sciences. The associated faculty are
all involved in research efforts that cross these boundaries, and can provide
a conduit between the CNS and the neuroscience research being conducted at
EU.
The Coulter Department combined with the EU Graduate Neuroscience Program
provides expertise in a number of areas of neuroscience. Already, the formation
of the GT Center for Nonlinear Science has begun to enhance interactions between
these bioscientists and bioengineers, and nonlinear dynamicists in the physics
and mathematics departments. The formation of the CNS will enhance that relationship
and facilitate the long-term development of this cross- disciplinary research
development.
The projects in this focus area fall into two categories. The first category
is the
nonlinear dynamics of small networks
of neurons. In particular,
this entails exploring the dynamics of individual cells and the coordination
of their dynamics in the formation of small neuronal circuits. The second
category deals with
large assemblages of neuronal elements
. Large networks
cannot be studied on a cell-by-cell basis, but instead are explored through
the analysis of the collective dynamics of populations of cells. A theme
which pervades all projects in both categories is the use of dynamical techniques
in conjunction with cutting-edge instrumentation to explore the behavior
of neurobiological systems. It is no exaggeration to say that we seek to
answer questions that could not be addressed to any serious extent without
this unique combination of tools.
Small Assemblies of Biological and Simulated Neurons:
Brain functions such as information processing, memory formation and motor
control often involve
oscillatory neuronal networks
. In investigating
the cellular mechanisms, biologists have frequently resorted to small networks,
typically from invertebrates. These networks are unique in that indivdual
cells with known properties are visually identifiable, and the nature of the
connectivity within such networks can often be determined on a cell-to-cell
basis. Detailed data permits the development of
biophysically-based models
to study the dynamics and function of such oscillating networks. Mathematical
modeling has been particularly useful in gaining insights into how such oscillations
are generated and how they contribute to nervous system function.
More recently, a method known as the dynamic current clamp technique permits
the interaction between real-time computational models and in-vitro cellular
experiments. An alternative technique is the development of analog very large-scale
integrated circuits (aVLSI) that possess dynamics similar to the computational
models. CNS participants have significant experience with both techniques
74
75
].
The objective of this theme of research is to investigate the use of
hybrid-systems
models as a tool to analyzing and understanding the dynamics of small neuronal
networks. Such hybrid systems are fundamentally experimental systems, yet
the real-time simulation component offers complete specificity of parameters
and complete observability of internal processess. Our efforts in this category
will focus on two fundamental issues:
mechanisms of half-center oscillations
and
mechanisms of spike-synchronization
. An underlying theme in both
lines of inquiry is to apply what is already know from a bifurcation analysis
of computational models to investigate the dynamics of the experimental system.
The leech heartbeat
central pattern generator
(CPG) is an oscillatory
neural network, which is one of the best described in terms of morphology,
connectivity and dynamics. The heartbeat CPG consists of a set of seven segmentally
repeated pairs of inhibitory heart (HN) interneurons located in the first
seven segmental ganglia. Such oscillators made up of mutually inhibitory
neurons or groups of neurons are often referred to as
half- center oscillators
76
]. A central challenge is to understand exactly
how synchronization between the individual elements of the oscillatory pair
occurs. To address this we will begin by improving existing dynamical models
so as to account for recent experimental results. The models assume that
the HN cells in isolation were not capable of rhythmic bursting oscillations
in the absence of synaptic coupling to other cells. We now know that isolated
HN cells are capable of rhythmic oscillations. A bifurcation analysis of
the model (see Fig.
) illustrates that the parameter
regime where individual cells are capable of intrinsically bursting is quite
small.
By incorporating real-time simulations with experiments on single HN neurons,
we will be able to modify HN cell conductances and test whether or not the
diagram above r3in
#1Bifurcation diagram of a single model HN cell showing different
dynamical oscillatory modes.
is a true representation of the dynamics of the
individual HN cells. We will then proceeed to study mechanisms of sychronization
between HN neurons, using both real-time simulations and are aVLSI models
of HN cells (see Fig.
a).
In a parallel project, the goal is to understand the fundamental dynamical
mechanism underlying spike synchronization.
Synchronization of spiking neurons
is
a fundamental process in both the generation of rhythmic patterns as well
as inforamation processing and representation. In the simplified case of
two coupled neurons, conventional wisdom has it that excitatory synapses,
as well as gap junctions (electrical coupling) lead to in-phase synchrony,
while inhibitory synapses lead to anti-phase synchrony. However, recent computational
and theoretical work has shown that such generalizations are questionable
in their generality. It has been demonstrated that inhibitory synaptic coupling
can lead to synchrony, excitatory coupling can be destabilizing, and the
gap junctional coupling does not always lead to synchrony and can cause anti-phase
synchrony to occur [
77
78
]. While experimental
evidence and computer simulations strongly support a role for inhibitory
synapses serving to sychronize neural networks, neither this phenomena nor
most of the others shown here have been conclusively demonstrated experimentally.
(One notable exception is ref. [
79
].)
Using our hybrid systems, we will couple pairs of neurons from the buccal
ganglia of the pond snail Helisoma Trivolvis. This animal was chosen since
the buccal ganglia is a symmetrical pair of clusters of neurons, this it is
possible to identify a pair of neurons with similar electrophysiological properties.
Our real-time simulation methodologies [
80
] enable
us to simulate the kinetics of fast synapses at rates up to 40 kHz. The first
step is to artificially couple pairs of neurons using this approach to study
how the dynamics is altered as the sign and kinetics of the coupling are
varied. We will also study the effect of electrical coupling.
(a)
(b)
Figure 6: a) aVLSI HN cell (top) coupled to in vitro
HN cell (bottom). b) In-Phase (top) and anti-phase (bottom) coupling between
a real (green) and simulated (blue) invertebrate motoneuron. Cross- correlations
shown to right.
The results of these experiments will yield in vitro bifurcation diagrams
that can be used to describe the parameter-coupling regimes of the experimental
system and attempt to validate earlier theoretical work on neuronal synchronization
with a real experimental system. The Fig.
b illustrates
data from Butera's laboratory, coupling a simulated spiking neuron running
in real-time with a single neuron from the pond snail. The top panel illustrates
in-phase synchrony between the in vitro and simulated neuron (cross-correlation
to right). The bottom panel demonstrates anti-phase synchrony under a different
set of synaptic parameters. Further efforts will aim to extend previous work
studying the syncronization dynamics of aVLSI neurons to hybrid systems of
aVLSI neurons and in vitro neurons.
Large Networks of Vertebrate Neurons:
We will
concentrate on two projects that involve very large collections of neuronal
elements. In one, the system consists of cultured networks of thousands of
mammalian neurons. In the other, the system is the human brain. These are
admittedly ambitious projects that are at the far frontier of what might
be possible. Nevertheless, they represent an important component of the CNS
scope in tacking new problems that require new methods as well as cross-disciplinary
thinking.
The first project will study the collective dynamics of ensembles of neurons
with
neural dynamical computing
in mind. Everything that mammalian
brains do, they do by the cooperation and interaction of many cells. We aim
to discover how information is represented and stored in a distributed fashion
in living neural networks, and how it is used to control behavior [
81
].
Most electrophysiology has been done using one or a few micropipette electrodes.
By recording neural activity optically, using voltage-sensitive dyes (or
fluorescent proteins), we will learn how activity propagates through the
excitable medium of a cultured network of several thousand mammalian neurons.
We designed and built a unique high-speed CCD camera for this purpose [
82
] (see Fig.
a). It represents
a significant advance in resolution over commonly used photodiode arrays,
and is unmatched in sensitivity by commercially-available high-speed imagers.
This allows us to image individual action potentials in many cells, without
averaging, which is necessary if we wish to observe the role of noise or
dynamics that vary from trial to trial. We developed a breakthrough in neural
culture [
83
] that has enabled us to grow
primary neuronal cultures for well over a year, on multi-electrode array
substrates (see Fig.
b). This allows us to record
spontaneous or elicited neural activity from 60 electrodes (a few hundred
cells) and to stimulate them non-destructively. We developed a feedback system
in which the network's own activity (motor output) influences its stimulation
(sensory input) via a simulated animal, the Neurally Controlled Animat [
84
]. With this tool, we have observed a wide variety
of chaotic and patterned dynamics in cultured networks, on time scales ranging
from milliseconds to months.
r3in
(a)
(b)
#1a) A high-speed camera
for neuronal tissue recordings. b) A multielectrode array for interfacing
to neuronal tissue in vitro.
A major bottleneck for this research is that we are
swamped with too much data, and insufficient analytical tools to find the
underlying structure in these patterns. This is precisely where close collaboration
with theorists in nonlinear dynamics may lead to advances in elucidating
the structure of the dynamical landscape of these nets, in multi-dimensional
activity space. And more relevant to human brains, how we can alter these
dynamics with feedback stimulation to model perception, learning, memory,
and other basic properties common to all neural systems. It is likely that
the meso-scale dynamics that we uncover with these new tools (both hardware
and theoretical) will be applicable to those in living animals, including
humans. By using a reduced preparation of a few thousand cultured neurons
(with glia), we have the unique ability to observe activity and changes in
every single cell that forms part of the net, to follow activity all the
way from the bottom up. And by using the net to control the behavior of simulated
(or mechanical) animals, we can demonstrate the relevance of our findings
to the top-down neuroscience community as well as utilize living neurons
in devices like biosensors and nonlinear biocomputers.
To understand neural circuits at this mesoscopic scale, and to make the
connection between individual cells and behavior, we will benefit from a
collaborative environment where theorists, modelers, and neurobiologists
work together toward a common goal. The proposed CNS will create just this
sort of environment.
Another project, this time involving
brain scans of human subjects
tackles the same fundamental problem: how can one recognize and usefully extract
patterns from extremely
large spatiotemporal data sets
? In a new collaborative
effort between EU and GT, we design, execute, and analyze experiments investigating
the relationship between brain activity and cognition [
85
]. The central question is how to understand the
large amounts of spatiotemporal data available to us in experiments that
use functional Magnetic Resonance Imaging. The technique allows one to get
both spatially and temporally resolved information about a subject's brain
activity. We are exploring new ways to analyze the large data sets using
recent ideas and techniques developed in nonlinear dynamics. In turn, we
are developing experimental protocols which make the best use of these methods'
strengths.
Experiments performed at EU School of Medicine monitor a subject's brain
activity while the subject performs a simple motor or cognitive task. Despite
ample evidence for the nonlinearity of neural activity [
86
], conventional neuroimaging studies take a linear,
subtractive approach to experimental design and analysis. Our approach takes
a different paradigm as its starting point, namely, that of bifurcation theory.
Thus, we seek situations under which a continuous variation of the cognitive
task results in a sudden qualitative change in a subject's behavior, an event
which we then view as a bifurcation of the neural system.
In addition to direct analysis of the spatiotemporal data by modern time
series methods, we will also develop a model-based approach to investigate
essential (if unglamorous) issues concerning the relationship between the
measured brain activity and the actual brain activity. This is needed to account
for unavoidable physiological factors affecting the measurements. For instance,
the relatively slow hemodynamic response smears out the measured activity
in a way which could mask intrinsically interesting complex dynamics.
Focus group: Stochastic dynamics in biological systems
The interplay of stochasticity and nonlinear dynamics leads to important
and often surprising results. The discovery and development of some of these
effects has been the focus of intense study among physicists. More recently,
attention has turned to the impact these ideas might have for biological
systems. Among the most famous of these phenomena are stochastic resonance
and Brownian ratchets [
87
].
CNS faculty has made primary contributions in this field, both in physical
and biological contexts. A primary function of CNS would be to sponsor an
interdisciplinary workshop and support visitors doing either collaborative
or closely related work. This would complement existing projects which are
funded individually. Such CNS sponsored interactions would create a fertile
environment and foster new lines of research. Our optimism stems in part
from the continuing success of the already functioning local program, the
EU-GT Biomedical Technology Research Center seed grant program. A number
of CNS faculty have entered collaborations in the life sciences with the
help of this program.
Our initial emphasis will be on three current areas of interest. The areas
are natural because they are important, are ripe for immediate progress, and
deal with topics on which we have considerable prior expertise.
The first concerns
rectified Brownian movement
. This phenomenon
potentially provides a unified mechanism for a great many basic cellular processes
whereby metabolic Gibbs free energy is converted into mechanical work [
88
]. The fundamental idea
is that in these processes ATP does not do mechanical work directly; rather,
it is responsible for switching on and off asymmetric boundary conditions
for thermal diffusion. In many instances, this mechanism explains what is
otherwise hypothesized to be some sort of direct chemomechanical conversion
of ATP into useful work. Recently, rectified Brownian movement has been used
to explain several molecular and cellular processes, including ubiquinone
transport across lipid membranes in electronic transport chains, allosteric
conformation changes in proteins, P-type ATPase ion transporters, rotary
arm enzyme complexes, the dynamics of actin-myosin cross bridges in muscle
fibers, and kinesin motion along microtubules.
As a result of these recent studies, there is reason hope that rectified
Brownian movement is a very general and in a sense universal mechanism in
the nanobiology of intracellular processes. The challenge now is to focus
on the more formidable examples, such as the interaction of myosin and actin
in muscle fibers, protein and RNA transport through membrane pores, and the
mechanism of bacterial flagella rotation.
In a second project, we explore how thermal noise
affects the mechanoelectrical transduction of hair cells in the auditory system.
In vitro experiments at EU by project collaborator F. Jaramillo (now at Carleton
College) showed that hair cells exhibit
stochastic resonance
, so that
the detection of weak signals was enhanced by the presence r3in
#1Illustration of a hair cell. For simplicity, only one hair bundle
and one ion channel is drawn (at the top and bottom, respectively). The inset
shows that the hair bundle is really an array of individual hairs, arranged
in ranks.
of added noise [
89
], see Fig.
. Stochastic
resonance has been observed in a variety of neuronal systems, and has led
researchers to ask whether it plays a functional role, for example in the
detection of predators or the identification of food [
90
]. To determine whether
noise plays a functional role is fraught with difficulties, of course. However,
the hair cell is a particularly apt candidate to explore this question, for
two reasons. First, since the transduction is mechanical the relevant source
of noise is easy to identify, namely Brownian motion of the hair bundle due
to the surrounding fluid. Second, the physiology of these cells, and particularly
the hair bundle, is well characterized, which makes it possible to write
down accurate equations of motion. Intriguingly, the hair cell shows stochastic
resonance at noise levels comparable to the inherent Brownian motion of hair
bundles in vivo.
There is room for progress on both theoretical and experimental sides.
The hair bundle is made up of many hairs within each rank, which raises the
interesting possibility that array enhanced effects may play a role. The
study of the full model, through numerical simulations at first, will allow
experiments to determine the expected
in vivo
behavior through a series
of
in vitro
runs where the bundle motion is constrained to move by
external manipulation, which has certain advantages over generating controlled
pressure waves in the fluid.
If confirmed, the hypothesis that noise plays a functional role in hair
cells could provide a neat explanation of a long-standing mystery: why are
the majority of hair cells in the cochlear not free-standing, but attached
to a rigid membrane? The issue is whether (as appears to be the case) noise
only enhances transduction for sub-threshold stimuli. If so, the function
of the free-standing population may be to detect the weakest signals, in which
case being unattached is beneficial since it means much larger Brownian fluctuations.
NIH is funding ongoing experiments at Carleton, and development of a model
of the hair bundle transduction process, in a collaboration between Jaramillo
and Wiesenfeld. In addition to workshop activity in this focus area, CNS
would support an extended visit of Jaramillo to GT; experimental resources
would be available in the laboratory of Ditto, who has long standing expertise
in the study of stochastic resonance.
The third project also involves close interactions between experiment and
theory, and concerns the role of
noise in neocortical interactions
Neocortical data obtained from experiments on rats [
91
] serve as a basis
for a biologically relevant mesoscopic neural network model. Exact results
can be obtained for noise driven binary interactions; the robustness of certain
dynamical properties allow us to extrapolate to more complex types of interactions.
This approach fills the gap between detailed biophysical simulations which
cannot make rigorous global predictions and generalized models which allow
exact statements but on a level of description remote from biology.
D  CNS Faculty
Director:
P. Cvitanovic', Glen Robinson Chair, Professor, Physics (COS)
Faculty:
J. Bellissard, Professor, Mathematics (COS)
L. A. Bunimovich, Regents' Professor, Mathematics (COS)
R. L. Calabrese, Professor, Biology (Emory University)
R.F. Fox, Regents' Professor, Chair of School of Physics (COS)
R. Grigoriev, Assistant Professor, Physics (COS)
R. Hernandez, Associate Professor, Co-director CCMST, Chemistry
(COS)
G.P. Neitzel, Professor, Mechanical Engineering (COE)
M. Schatz, Assistant Professor, Physics (COS)
A. Shilnikov, Assistant Professor, Mathematics (Georgia State University)
T. Uzer, Professor, Physics (COS)
E.R. Weeks, Assistant Professor, (Emory University)
K. Wiesenfeld, Professor, Physics (COS)
Associate Faculty:
C. Aidun, Professor,
Mechanical Engineering
(COE)
G. Bao, Associate Professo, Biomedical Engineering (COE)
F. Bonetto, Assistant Professor, Mathematics (COS)
R. J. Butera, Jr., Assistant Professor, Electrical and Computer
Eng. (COE)
Y-H. Chen,     , Mechanical Engineering (COE)
Abbreviations:
GT  Georgia Institute of Technology
EU  Emory University
EUSM  Emory University School of Medicine
GSU  Georgia State University
COS  College of Sciences
COE  College of Engineering
BME  Georgia Tech/Emory Biomedical Engineering Department
ECE  Department of Electrical & Computer Engineering
ME  Mechanical Engineering
Chem  School of Chemistry and Biochemistry
Math  School of Mathematics
Phys  School of Physics
CCMST  Center for Computational Molecular Science &
Technology
CDSNS  Center for Dynamical Systems & Nonlinear Studies
3  Education, Human Resources, Diversity,
and Outreach
A unique aspect of nonlinear science is that research
is driven by the flux of ideas across disciplinary boundaries. Given the essential
role students at all levels play in conducting research, such flux is most
effectively generated through the establishment of a
cross-disciplinary
training program
integrating research, education, and training with the
dual goals of attracting a wide range of undergraduate, graduate, and postdoctoral
students to physics and providing them with a stimulating and thoroughly
modern learning environment. To ensure the success of the program, a large
fraction of the CNS funded graduate students will be co-advised, with one
advisor from student's home department and the other external, at least one
of the advisors being a CNS member. This arrangement will greatly assist
in initiating new cross-disciplinary collaborations.
On the other hand, equipping students with skills that are cross-disciplinary,
method based, rather than discipline specific, nonlinear science is uniquely
positioned to
offer the students a broad, diverse education
, and prepare
them for today's rapidly evolving professional environment. Coupled with
the cross-disciplinary nature of CNS, such training will meet the recommendation
of the NRC report [
] to physics departments ``to review and revise their
curricula to ensure that they are engaging and effective for a wide range
of students and that they make connections to other important areas of science
and technology''. In order to effectively achieve these goals, the proposed
research and training programs will transcend departmental boundaries, differing
from the conventional training model in a number of significant ways.
The success of the training program will crucially depend on the availability
of the infrastructure that can only be created by a Center such as the proposed
CNS. The need for a strong foundation in the analysis of nonlinear dynamical
systems is common to many research programs in science, engineering, and
mathematics. At GT, courses on nonlinear dynamics [
92
] are currently taught
in the mathematics and physics departments, each from a different perspective,
and draw students from biomedical, chemical, aerospace, civil, electrical
and mechanical engineering, materials science, and chemistry. While students
would clearly benefit from a program based on a broad, coherent, and unified
view of nonlinear science, college and school boundaries need to be superseded
by a framework that integrates what otherwise would be isolated collaborations
between individual faculty members. The program built around the CNS initiative
will generate a new level of integration in graduate and post-graduate education
by involving faculty in different departments in an effort to unify the nonlinear
science curriculum across the participating schools. Equally important, combining
the resources of the Center's MRCs will make it possible to offer undergraduate
level cross-disciplinary courses to students from participating institutions.
The core of the cross-disciplinary training program will consist of an
Introductory Nonlinear Science
course on the mathematical and computational
techniques of nonlinear science followed by a rotating sequence of
Applications
of Nonlinear Dynamics
courses based on the research interests of individual
Center members and rounded out by a semester of
Special Topics in Nonlinear
Science
featuring research on specific projects carried out by small
teams supervised by faculty members with complementary expertise and perspectives.
The level of courses will vary from advanced undergraduate to second year
graduate, targeting the range of students that can be most effectively recruited
into research programs. As one of the new initiatives, CNS will offer an
advanced web-based year-long gradute nonlinear course [
], which will play an important role in reaching
and educating students at remote locations and attracting them to the program.
The courses will gather all nonlinear science students in an activity that
stresses commonalities among various fields, and it will provide a sense
of intellectual community fundamental to the success of the program. The
course work will require coordinated collaboration with peers and teachers
from different backgrounds, helping students develop communication skills
that will prove invaluable in future careers in industry or academia.
This training will equip young researchers with the tools and intuition
needed to tackle complex nonlinear problems arising in many guises and various
technical fields. In order to offer students a deep learning experience outside
the Ph.D. thesis research, the educational core of the training program will
be supplemented by the following important components. A
welcoming workshop/retreat
before the start of each fall semester, will introduce the incoming young
researchers to the CNS. New students will pair up with a nonlinear science
advisor, preferrably from outside their own department, who will oversee
the student's progress during the first two years in tandem with the departmental
advisor. An
Interdisciplinary Nonlinear Science Seminar
series [
92
], initiated in January
2001, has drawn a wide attendance from the participating departments and
institutions. CNS would enable us to run the seminar on an ongoing basis
and to broaden its scope to engineering and biological applications.
Graduate
Student Seminars
organized for and by the students will give them opportunity
for public presentation of their own research in a supportive setting, engage
them in mutually beneficial exchange of ideas, and generally enhance their
communication skills; it will also provide a forum for directed discussions
on ethics and conflict of interests in research. An active
visitor program
will help promote and maintain national, international, and industrial collaborations.
Priority will be given to visitors whose research relates to that of participating
CNS faculty, and who demonstrate potential as external mentors to our graduate
students and junior researchers. A
Regional Nonlinear Workshop
, a
yearly cross-disciplinary meeting organized jointly with other Southeastern
universities such as Duke, U. of Alabama, and U. of Florida, will expose
students from regional universities to the forefront of research, and give
them an opportunity to present their own work in poster sessions.
Joe Ford Fellowships
, named in honor of late
GT Professor Joseph Ford, one of the pioneers of classical and quantum chaos,
will sponsor several outstanding postdoctoral fellows each year. These Fellows,
associated with the Center, rather than its individual members, will have
complete freedom in choosing their research directions and serve as a glue
between different research projects. In addition, by combining the resources
of the Center, the Departments, and the Institute, the CNS will be able to
make competitive
graduate fellowship
offers to incoming graduates, making
the program more attractive to a wider range of students. These fellowships
will be primarily used to support co-advised students, covering the first
two years during which students are normally supported as teaching assistants,
allowing them to concentrate on the study and research in nonlinear science.
Joining the forces of the affiliated faculty and students is instrumental
in making a coherent contribution to a number of outreach programs organized
under the umbrella of the GT Center for Education Intergrating Science, Mathematics,
and Computing (CEISMC) [
93
] and targeted at the middle and
high school students and teachers. For instance, participation in such programs
as
FutureScape
and the NSF-sponsored
Integrating Gender Equity And
Reform
(InGEAR) will help educate girls and young women about careers
in nonlinear science, broaden their horizons and raise their aspirations
by exposing them to careers they may not have contemplated otherwise. The
Center will also actively participate in the new
Post-secondary Readiness
Enrichment Program
(PREP) created by the University System of Georgia.
This program addresses primarily the seventh grade and is designed to assist
students and their parents in making timely, informed decisions that will
adequately prepare young people for their higher education and career goals.
The Center will also provide the foundation for continuing educating in the
area of Nonlinear Science of those high school and middle school science
and mathematics teachers enrolled in the year-round
Georgia Industrial
Fellowships for Teachers
(GIFT) program, whose primary goal is to provide
the exposure to the cutting edge research in the academic and industrial
settings.
At the college level, following up on the success of the NSF Southeast
Applied Analysis Center [
] outreach program, CNS members will be
sent to deliver lectures on various topics of modern physics, biology and
mathematics as
Center Ambassadors
to Southeast region non-research
educational institutions and historically black colleges and universities.
CNS will offer teaching reduction to the members who perform exceptionally
as such lecturers, and implement a series of measures to aid recruitment
and retention of students from under-represented groups, such as offering
Summer
Internships for Minority Undergraduates
Both GT Physics and GT Chemistry run highly
successful NSF-funded
Research Experience for Undergraduates
(REU)
programs, bringing some 30 gifted undergraduates to GT every summer, some
of which are advised by the CNS faculty. In summary, the goals of the CNS
outreach program are to bring the inspiration of cutting edge physics research
to bright undergraduates, K-12 students, and teachers, to build bridges to
faculty isolated at non-research institutions, and to inform talented students
about the attractiveness of GT graduate programs.
The above range of activities, while highly desirable, cannot be sustained
by the individual grants. The very existence of the proposed program hinges
on the availability of cross-disciplinary funding, something that only CNS
can make possible.
4  Shared Facilities
The CNS will require substantial single- and parallel-processor
computing resources to implement the MRCs' objectives. GT's CCMST [
94
],
whose co-director, R. Hernandez, is a CNS member, is presently providing a
limited number of compute cycles on a 72-processor IBM SP2 to CNS members.
Up to one third of this machine will be available to the CNS with the part-time
funding of a research scientist and a commitment to CNS to upgrade part of
the facility at year 3.
5  Collaboration with Other Sectors
Cross-disciplinary workshops, internships and other training initiatives
will make use of the extensive connections the CNS faculty has with other
institutions and labs. We now provide a few examples of such cross-fertilization
initiatives already in place. First, the already established
institutional
cross-connections
CNS will collaborate intensively with the
NSF IGERT
programs
at Northwestern [
], Cornell and U. of Arizona, as
well as with a number of other leading centers for nonlinear science (check
the Center webpage [
95
] for a complete list).
The ties are particularly close to the Northwestern IGERT, originally led
by P. Cvitanovic'.
GT is one of the six university partners in the team which operates
the
Oak Ridge National Laboratory
(ORNL), linking the lab to academic
research. Geographically close, and with common interests in complex and nonlinear
systems, GT CNS and ORNL already plan several collaborative initiatives (detailed
below).
As a member of the External Advisory Board for Space Medicine and Life
Sciences Research Center, GPN advises researchers at the
Morehouse School
of Medicine
on fluid mechanics aspects of tissue growth.
The CNS fluid dynamics GT Physics & ME program [MS,RG,GPN] interacts
the
Institute of Paper Sciences & Technology
(C. Aidun) housed
at GT, where the fundamental problem of pattern formation is relevant to increased
efficiency in coating of paper.
GSB and KW have formed a ``Hyperscanning Consortium'' with colleagues
at Baylor College of Medicine, Princeton University and Caltech, in order
to perform synchronized MR scanning of people interacting with each other,
and study brain patterns of interacting groups of humans.
Every CNS faculty member has extensive external collaborations. Here we highlight
a few examples of
cross-institutional research ties
that would be
enhanced by the CNS visitor, internship and workshop programs:
KW and R. York (UC Santa Barbara, Electrical Eng.), T. Heath (GT Research
Inst. staff scientist) -
Fast Dynamical Control of Antenna Arrays
: Experiments
at Santa Barbara are testing a new scheme for very fast manipulation of electromagnetic
beams, scheme based on GT theoretical advances in understanding how synchronized
nonlinear oscillators behave under variations of the element parameters.
ERW and H. Stone, S. Koehler, and S. Hilgenfeld (Harvard) -
Drainage
flow through channels in foams
: Changing the surfactant properties changes
the flow from plug-like to Poiseuille-like inside the channels.
KW and F. Jaramillo (Carleton College, Biology) -
The role of noise
in the auditory system
: The ``stochastic resonance'' experiments are being
performed at Carleton College, and a model for the hair cell is under development
at GT (see p.
pageref
).
Internships:
We have contacted a number of potential hosts who have
expressed their interest in participating in the internship program (see p.
pageref
). Close contacts between the interns' advisors
and hosts will contribute substantially to turning internships into full-fledged
collaborations. To give a flavor of what we have in mind, we give here two
examples of potential hosts and what they would offer. Others will be contacted
on case by case basis, pending specific CNS student or host research initiatives.
D. del-Castillo-Negrete, Oak Ridge National Laboratory.
Theorist,
expert on the application of dynamical systems methods to the study of transport
problems in fluids and plasmas, dynamics of many-body Hamiltonian systems
with long range interactions, and pattern formation in reaction-diffusion
systems, research interests shared with the CNS faculty. GT and ORNL are already
partners on the institutional level, with long experience of GT students working
at ORNL.
R.E. Ecke and R. Mainieri, Los Alamos National Laboratory.
Experiments
on pattern formation in fluids. Theory of nonlinear dynamics, statistical
mechanics of chaotic systems.
An internship at ORNL and LANL will provide a student with a unique opportunity
to explore the research environment at a National Laboratory.
Workshops:
CNS plans a series of workshops. On the national level,
CNS will propose to host the
Dynamics Days 2005
, the main US annual
meeting in nonlinear science, possibly in collaboration with the Oak Ridge
National Lab. The
Regional Nonlinear Workshop
(see p.
pageref
) currently under preparation is
ORNL-CNS Interdisciplinary Workshop on Transport
DATE: June 2003
LOCATION: Oak Ridge National Laboratory
ORGANIZERS: D. del-Castillo-Negrete (ORNL) and P. Cvitanovic' (CNS)
FUNDING: Shared ORNL/CNS, with emphasis on graduate student stipends. 40
participants
PROCEEDINGS: A focus issue of the CHAOS journal
SPEAKERS:
(preliminary list) C. Jones (Brown, transport in oceanographic
flows), R. Pierrehumbert (U. Chicago, transport in atmospheric flows), T.
Tel (Eotvos U., Hungary, transport in chemical active flows), D. Astumian
(U. of Maine, transport in biological systems), S. Wiggins (Bristol U., UK,
transport in dynamical systems), R. Behringer (Duke, transport in granular
flows), J. Gollub (Haveford College, experiments on transport), G. Zaslavsky
(Courant Institute, NYU, transport in plasmas).
ABSTRACT: The study of transport is an important problem of common interest
to many areas of science and technology including plasma physics, biology,
chemistry, oceanography, atmospheric sciences, engineering, and dynamical
systems. The goal of this cross-disciplinary workshop is to bring together
experts from these areas for a discussion of nonlinear dynamics and complex
systems techniques in the study of transport.
CNS academic environment
: No less important than ties to other institutions
is the immediate context in which CNS will operate. The overriding concern
in setting up the Center structure is ensuring sufficient fluidity in faculty
composition to enable us to recruit new talent into CNS as new cross-disciplinary
research directions open up. GT and EU provide rich intellectual environment
which will greatly enhance CNS effectiveness as a platform for cross-disciplinary
interactions. GT School of Mathematics CDSNS [
96
] contributes
excellent visiting researchers, seminars, and training opportunities in mathematical
methods for nonlinear science. To name but a few, preeminent scientists such
as E. Carlen, M. Loss, L. Erdos, E. Harrell (GT mathematical physics group),
K. Mischaikow (GT Mathematics / director CDSNS), W. Gangbo (GT Mathematics
/ CDSNS), M. Borodovsky (GT Biomathematics), D. Dusenbery, R. Wartel (GT
Biology), G. Hentschel and F. Family (EU Physics), and N. Chernov and N.
Simanyi (U. Alabama Mathematics, Birmingham) will interact with CNS faculty
and visitors. These excellent researchers could have equally well already
been listed as CNS faculty, and they and others in Atlanta area will be inducted
if new research directions merit their joining CNS.
6  International Collaboration
CNS plans to co-organize a number of international workshops. P. Cvitanovic'
is the
Secretary of the European Dynamics Days Governing Board
, a
Honorary
Chair of the Let's Face Chaos
conference (Maribor, Slovenia May 2002),
and a member of other international conference committees. Since 1981 he
has organized or co-organized more than 25 conferences, workshops and schools
in Europe and US, and has a well established international partner network.
Internationally, some fifteen
``Sister Nonlinear Science Centers"
located
in Mexico, Germany, Denmark, United Kingdom, Italy, Israel, Argentina, Hungary,
Chile and Austria are CNS's potential partners in organizing workshops, exchanging
researchers, and hosting interns (for a complete list, please check the Center
webpage [
97
]). A few examples:
Center for Chaos and Turbulence Studies
] - M.H. Jensen, director.
A leading European center in physics of complex systems ranging from neuronal
activity to quantum chaos, housed at the Niels Bohr Institute, Danish Technical
University, and Nordita, Copenhagen. CATS will host CNS researchers and interns.
CNS will host the Danish Research Academy
Danish Ph.D. School in Nonlinear
Science
supported students and visitors. As many European Ph.D. programs
require research training abroad, these and other similar exchanges will
be very beneficial to the overall CNS recruiting effort.
Centro Internacional de Ciencias
, Cuernavaca, Mexico - T.H.
Seligman, director. Promotes scientific exchange, with particularly strong
Latin American visitor and conference program. CNS and CIC have a tentative
agreement to co-organize two ``pan-American'' workshops, the first one on
``Classical and quantum chaos in few- and many-body systems'' in winter 2004
and the the second one in winter of 2006.
Max Planck Institute for the Physics of Complex Systems
, Dresden
- H. Kantz, Head, Nonlinear dynamics and time series analysis group. The
main European meeting ground for young researchers and leading international
scientists in physics of complex systems, with intensive workshop, visitor,
postdoctoral and exchange program. Collaboration with CNS desired in order
to strengthen ties to US.
In addition - the space allotted does not allow a detailed listing - each
individual CNS faculty member maintains
international research ties
that will enhance and be enhanced by the CNS visitor, internship and workshop
programs.
7  Seed Funding and Emerging Areas
The unique position of nonlinear science lying on the interface of two
or three different disciplines is reflected in the funding patterns specific
to cross-disciplinary research: although established research programs have
somewhat broader funding opportunities, it is substantially more difficult
to find individual funding opportunitities for new research directions, since
typically they will not be given priority by either of the core disciplines.
This puts special emphasis on the availability of
seed funding in emerging
areas
, especially high-risk ones.
The CNS will be able to provide such funding quickly and in an extremely
flexible way. The highest priority will be given to collaborative research
lying on the interface of different disciplines and involving faculty from
different departments. Second priority will be assigned to joint research
by the faculty members in the same department or discipline. In both cases
at least one of the investigators will come from the CNS faculty. In general,
the goal of seed funding will be to initiate and stimulate preliminary research
in the novel and promising areas of opportunity, where no existing patterns
of funding exist. Once the potential of a given direction is proven at the
preliminary stage, individual and/or group funding from more conventional
sources will be sought.
The availability of seed funds will be especially critical for the success
of
junior faculty members
who typically do not have other sources of
funding which can be used to initiate new research directions. Given the
difficulties that even senior faculty members face in funding new research
directions, this task becomes especially and unnecessarily complicated for
junior faculty, who do not yet have a track record of successful research
to warrant favorable consideration by the granting agencies. Providing junior
faculty with such flexible funding will save a lot of time required otherwise
to prepare, submit, and undergo a review of several full-scale exploratory
proposals, enabling the junior faculty to concentrate instead on the proposed
research. All of this is especially true where nonlinear science with its
cross-disciplinary focus is concerned.
As was repeatedly stressed throughout this proposal, a distinguished feature
of nonlinear science is that it is driven by cross-pollination of ideas between
different disciplines as well as between different fields within natural
sciences, engineering and medicine. As a means to promote such cross-pollination,
CNS will organize and host a series of bi-annual
Applied Nonlinear Science
Workshops
, bringing together CNS faculty and students and researchers
from regional universities, national laboratories, and industrial research
centers. Each year the workshops will target a different application area
(e.g., neural dynamics, control of extended systems, plasma physics, biological
excitable systems, etc.), fostering collaborations and transfer of ideas,
creating new approaches, and stimulating new research directions by bringing
together specialists with complementary research expertise.
Organization of such workshops depends crucially on the availability of
centralized funding and infrastructure. On the other hand, by limiting the
scope of the meeting and bringing together a sharply focused group of researchers,
such meetings will play a role that neither the individual contacts nor the
very broad international conferences such as the SIAM Snowbird meeting play.
The workshops will serve the dual purposes of disseminating the methodology
developed by the nonlinear community to other fields with the latter providing
the focus for the research directions in the nonlinear science and, at the
same time, establishing new collaborations between members of different institutions.
Complementing the above research-oriented activities will be a range of
education-oriented activities, such as a Regional Nonlinear Workshop, internship
support for outstanding students, and the development of a web-based nonlinear
courses, all of which are expected to have a strong impact on the research
component.
The
Regional Nonlinear Workshop
will
differ significantly from the Applied Nonlinear Science Workshop in that
it will be oriented primarily towards students of all levels. Scheduled yearly,
this cross-disciplinary meeting will be organized jointly with other Southeastern
universities such as Duke, U. of Alabama, and U. of Florida, it will be hosted
on an alternating basis by the CNS and the fellow institutions. The main
goal of the workshop will be to expose students from regional universities
to the forefront of research, and give them an opportunity to present their
own work in poster sessions. This workshop will have a broader scope, encompassing
the whole field of nonlinear science. This broad scope will give the students
the sense of diversity and at the same time closeness of the nonlinear science
community, highlighting the many existing and prospective links between different
directions pursued by different research groups.
CNS will offer to students and post-doctoral fellows
of exceptional promise (either from CNS, or from outside)
internship support
the goal of which is to significantly broaden and strengthen their education
and provide young researchers with additional cross-disciplinary perspectives
by exposing them to different experimental, computational, and theoretical
approaches, using the expertise not available at GT/EU. For example, a trainee
working on theoretical projects in the home institution might intern in an
experimental lab in a host institute, or
vice versa
. In other cases
the internship would provide a specific experimental technique or theoretical
approach not readily available at the home institution, but important for
the trainee's thesis research. In addition to this, the internship program
will play an instrumental role in forging new collaborations and finding
new areas of opportunity.
The trainee, aided by the home institution advisor, will make the initial
contact with the internship advisor. A brief proposal will then be written
that clearly states the goals of the internship, which is expected to last
3-6 months and possibly form the basis for a publication. The internships
will be reviewed after the first three months by the CNS and home institution
advisors to determine whether progress is sufficient to warrant an extension.
The trainee will present his/her results in the Graduate Seminar Series, and,
upon returning to the home institution, in the form of a written report.
Yet another component of the educational activity of the center, and an
integral part of its outreach program, a
web-based nonlinear course
will play an important role in reaching and attracting students from remote
locations, providing them with the inside look at the current research topics
in classical and quantum chaos. Such course could be based on the ChaosBook
], a hyperlinked web-based course currently under
design by P. Cvitanovic' and coauthors, a novel and unique approach to teaching
an advanced graduate level nonliner physics course. In addition to the extremely
valuable potential of accessing the broadest possible audience, the major
advantage of a web-based course compared to a textbook-based course is a
unique opportunity to incorporate the dynamical aspects into the presentation.
Such dynamical aspects realized through animations, java applets, interactive
scripts, and so on, can only be implemented with a considerable investment
of time and resources. Given the scope of the material, this will require
involvement of a group of advanced undergraduate and/or graduate students
for an extended period of time. Although clearly innovative and beneficial
for the whole nonlinear science community, such a project cannot be supported
by any individual grant and requires a centralized approach.
References
[1]
``Physics in a new era: an overview,''
National
Research Council report
(National Academy Press, Washington, D.C., 2001);
www.nap.edu/books/0309073421/html/
[2]
Center for Nonlinear Science,
www.cns.gatech.edu
[3]
Northwestern U. Dynamics of Complex Systems in Science
and Engineering,
www.complex-systems.northwestern.edu
[4]
Centre for Chaos And Turbulence Studies,
www.nbi.dk/CATS/Welcome.html
[5]
The Southeast Applied Analysis Center,
www.math.gatech.edu/saac
[6]
P. Cvitanovi\'c
et al.
Classical and Quantum
Chaos
, advanced graduate course, available electronically at
ChaosBook.org
[7]
A. Wirzba, N. Sondergaard, and P. Cvitanovi\'c,
Wave
Chaos in Elastodynamic Cavity Scattering
, submitted to
Phys. Rev.
Lett.
[8]
M. C. Gutzwiller,
Chaos in Classical and Quantum
Mechanics,
(Springer-Verlag, New York, 1990).
[9]
T. Uzer,``Rydberg electrons in crossed fields: A paradigm
for nonlinear dynamics beyond two degrees of freedom,''
Physica Scripta,
T90
, 176 (2001)
[10]
M. R. W. Bellerman, P. M. Koch, and D. Richards, ``Resonant,
elliptical-polarization control of e microwave ionization of hydrogen atoms,''
Phys. Rev. Lett.
78
, 3840 (1997).
[11]
P. M. Koch, E. J. Galvez, and S. A. Zelazny, ``Beyond
one- dimensional dynamics in the microwave ionization of excited H atoms:
surprises from experiments with collinear static and linearly polarized electric
fields,''
Physica D
131
, 90 (1999).
[12]
E. Oks and T. Uzer, ``New Guiding Principles for Ionization
Yield in Rotating Microwave Fields,''
J. Phys. B: At. Mol. Phys.
33
L533-538 (2000).
[13]
S. Mallat,
A wavelet tour of signal processing,
(Academic Press, San Diego, 1999).
[14]
C. Jaffé, D. Farrelly, and T. Uzer, ``Transition
State Theory without Time-Reversal Symmetry: Chaotic Ionization of the Hydrogen
Atom,''
Phys. Rev. Lett.
84
, 610-613 (2000).
[15]
S. Wiggins, L. Wiesenfeld, C. Jaffé, and T.
Uzer, ``Impenetrable barriers in phase-space,''
Phys. Rev. Lett.
86
5478-5481 (2001).
[16]
S. Wiggins,
Chaotic Transport in Dynamical Systems
(Springer-Verlag, New York, 1992).
[17]
S. Wiggins,
Normally Hyperbolic Invariant Manifolds
in Dynamical Systems
(Springer-Verlag, New York, 1994).
[18]
W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross,
``Heteroclinic connections between periodic orbits and resonance transitions
in celestical mechanics,'' CHAOS
10
, 427 (2000).
[19]
L. A. Bunimovich and J. Rehacek, ``Nowhere dispersing
3d billiards with nonvanishing Lyapunov exponents,"
Comm. Math. Phys.
189
729 (1997).
[20]
L. A. Bunimovich and J. Rehacek, ``How high-dimensional
stadia look like,"
Comm. Math. Phys.
197
, (1998)
[21]
L. A. Bunimovich and J. Rehacek, ``On the ergodicity
of high-dimensional focusing billiards,''
Ann. Inst. H. Poincare
68
(1998)
[22]
T. Papenbrock, ``Lyapunov exponents and Kolmogorov-Sinai
entropy for a high-dimensional convex billiard,"
Phys. Rev. E
61
1337 (2000).
[23]
T. Papenbrock, T. Prosen, ``Quantization of a billiard
model for interacting particles,"
Phys. Rev. Lett.
84
, 262 (2000).
[24]
L. A. Bunimovich, ``Mushrooms and other billiards with
divided phase space,"
Chaos
11
, 1 (2001).
[25]
M. G. Raizen, J. Koga, B. Sundarum, Y. Kishimoto, H.
Takuma and T. Tajima. ``Stochastic cooling of atoms using lasers,"
Phys.
Rev.
A58
4757 (1998).
[26]
E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield,
and D. A. Weitz, ``Three-dimensional direct imaging of structural relaxation
near the colloidal glass transition,''
Science
287
, 627-631
(2000).
[27]
E. R. Weeks and D. A. Weitz, ``Properties of cage rearrangements
observed near the colloidal glass transition,'' submitted to
Phys. Rev.
Lett.
, preprint:
cond-mat/0107279
(2001).
[28]
R. Hernandez and F.L. Somer, ``Stochastic Dynamics
in Irreversible Nonequilibrium Environments. 1. The Fluctuation-Dissipation
Relation,"
J. Phys. Chem.
B 103
, 1064 (1999).
[29]
T. Shepherd and R. Hernandez, ``Chemical Reaction Dynamics
with Stochastic Potentials Beyond the High-Friction Limit,"
J. Chem. Phys.
115
, 2430 (2001).
[30]
``Collision Based Computing,'' Special Issue of the
Journal
Multiple Valued Logic
, No.5-6 (2001).
[31]
L. A. Bunimovich, ``Walks in Rigid Environments,"
Physica
A279
169 (2000).
[32]
L. A. Bunimovich, ``Walks in Rigid Environments: Symmetry
and Dynamics,"
Asterisque
(2002)
[33]
L. A. Bunimovich and M. Khlabystova, ``Localization
and Propagation in Random Lattices,"
J. Stat. Phys.
104
1155
(2001).
[34]
A. S. Shilnikov and G. Tsimbaluk, ``Parabolic spikes
through the Blue Sky Catastrophe in a neuron model,'' in preparation.
[35]
R. Artuso, E. Aurell and P. Cvitanovi\'c, ``Recycling
of strange sets I: Cycle expansions,''
Nonlinearity
, 325 (1990).
[36]
P. Cvitanovi\'c and B. Eckhardt, ``Periodic-Orbit Quantization
of Chaotic Systems,''
Phys. Rev. Lett.
63
, 823 (1989).
[37]
R.F. Fox and M.H. Choi, ``Generalized Coherent States
and Quantum-Classical Correspondence,''
Phys. Rev.
A 61
, 032107
(2000).
[38]
J. Geronimo, G.C. Donovan and D.P. Hardin, ``Intertwining
multiresolution analysis and the construction of piecewise polynomial wavelets,''
SIAM J. Math. Anal.
27
, 1791 (1996).
[39]
D. Quéré, ``Fluid coating on a fiber,''
Annu. Rev. Fluid Mech.
31
, 347 (1999).
[40]
D. R. Bassett, ``Hydrophobic coatings from emulsion
polymers,''
J. Coat. Technol.
73
, 43 (2001).
[41]
P. Hinz and H. Dislich, ``Anti-reflecting light-scattering
coatings via the sol-gel-procedure,''
J. Non. Cryst. Solids
82
411 (1986).
[42]
A. Kitamura, ``Thermal effects on liquid film flow
during spin coating,''
Phys. Fluids
13
, 2788 (2001).
[43]
H. E. Huppert, ``Flow and instability of a viscous
current down a slope,''
Nature
300
, 427-429 (1982).
[44]
A. L. Bertozzi and M. P. Brenner, ``Linear stability
and transient growth in driven contact lines,''
Phys. Fluids
530-539 (1997).
[45]
R. O. Grigoriev, ``Control of Evaporatively Driven
Instabilities of Thin Liquid Films,'' submitted to
Phys. Fluids
[46]
D. E. Kataoka, S. M. Troian, ``A theoretical study
of instabilities at the advancing front of thermally driven coating films,''
J. Colloid and Int. Sci.
192
, 350-362 (1997).
[47]
R. O. Grigoriev,
unpublished.
[48]
D. P. Birnie, ``Rational solvent selection strategies
to combat striation formation during spin coating of thin films'',
J.
Mater. Res.
16
, 1145 (2001).
[49]
S. J. VanHook, M. F. Schatz, J. B. Swift, W. D. McCormick,
and H. L. Swinney, ``Long-wavelength surface-tension-driven Bénard
convection: experiment and theory,''
J. Fluid Mech.
345
, 45
(1997).
[50]
M. F. Schatz and G. P. Neitzel, ``Experiments on thermocapillary
instabilities,''
Annu. Rev. Fluid Mech.
33
, 93 (2001).
[51]
R. J. Riley, G. P. Neitzel, ``Instability of thermocapillary-buoyancy
convection in shallow layers. Part 1. Characterization of steady and oscillatory
instabilities,''
J. Fluid Mech.
359
, 143-164 (1998).
[52]
D. Semwogerere and M. F. Schatz, ``Evolution of hexagonal
patterns from controlled initial conditions in a Bénard-Marangoni convection
experiment,"
Phys. Rev. Lett.
88
(available online-in print
Feb 4, 2002).
[53]
L. B. S. Sumner, G. P. Neitzel, J. P. Fontaine, P.
Dell'Aversana, ``Oscillatory thermocapillary convection in liquid bridges
with highly deformed free surfaces: Experiments and energy-stability analysis,''
Phys. Fluids
13
, 107-120 (2001).
[54]
V. Petrov, M. F. Schatz, K. A. Muehlner, S. J. VanHook,
W. D. McCormick, J. B. Swift, H. L. Swinney, ``Nonlinear control of remote
unstable states in a liquid bridge convection experiment,''
Phys. Rev.
Lett.
77
, 3779-3782 (1996).
[55]
S. Benz, P. Hintz, R. J. Riley, and G. P. Neitzel,
``Instability of thermocapillary-buoyancy convection in shallow layers. 2.
Suppression of hydrothermal waves,''
J. Fluid Mech.
359
, 165-180
(1998).
[56]
G. M. Whitesides and A. D. Stroock, ``Flexible methods
for microfluidics,''
Phys. Today
54
, 42 (2001).
[57]
A. Oron, S. H. Davis, and S. G. Bankoff, ``Long-scale
evolution of thin liquid films,"
Rev. Mod. Phys.
69
, 931-980
(1997).
[58]
H. Aref, ``Chaotic advection of fluid particles,''
Phil. Trans. R. Soc. Lond. A
333
, 273-281, (1991).
[59]
J. M. Ottino, ``Mixing, chaotic advection, and turbulence,''
Annu. Rev. Fluid Mech.
22
, 207 (1990).
[60]
D. Rothstein, E. Henry, and J.P. Gollub, ``Persistent
Patterns in Transient Chaotic Fluid Mixing,''
Nature
401
, 770-772
(1999).
[61]
R. H. Liu, M. A. Stremler, K. V. Sharp, M. G. Olsen,
J. G. Santiago, R. J. Adrian, H. Aref, and D. J. Beebe, ``Passive mixing
in a three-dimensional serpentine microchannel,''
J. MEMS
190-197 (2000).
[62]
F. X. Witkowski, K. M. Kavanagh, P. A. Penkoske, R.
Plonsey, M. L. Spano, W. L. Ditto, and D. T. Kaplan, ``Evidence for determinism
in ventricular fibrillation,''
Phys. Rev. Lett.
75
, 1230-1233
(1995).
[63]
F. X. Witkowski, L. J. Leon, P. A. Penkoske, W. R.
Giles, M. L. Spano, W. L. Ditto, and A. T. Winfree, ``Spatiotemporal evolution
of ventricular fibrillation,''
Nature
392
, 78-82 (1998).
[64]
P. J. Holmes, J. L. Lumley, and G. Berkooz, ``Turbulence,
Coherent Structures, Dynamical Systems and Symmetry'' (Cambridge University
Press, Cambridge, 1996).
[65]
C.W. Rowley, J.E. Marsden, ``Reconstruction equations
and the Karhunen-Loeve expansion for systems with symmetry,''
Physica
142
, 1 (2000).
[66]
F. Qin, E. E. Wol f, H.-C. Chang, ``Controlling spatiotemporal
patterns on a catalytic wafer,''
Phys. Rev. Lett.
72
, 1459 (1994).
[67]
S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez,
I. G. Kevrekidis, E. S. Titi, and T. J. Mountziaris, ``Order reduction for
nonlinear dynamic models of distributed reacting systems,''
J. Process
Contr.
10
, 177-184 (2000).
[68]
F. Christiansen, P. Cvitanovi\'c, and V. Putkaradze.
``Spatiotemporal chaos in terms of unstable recurrent patterns,''
Nonlinearity
10
, 1 (1997).
[69]
S.M. Zoldi and H.S. Greenside, ``Spatially localized
unstable periodic orbits of a high-dimensional chaotic system'',
Phys.
Rev.
E 57
(1998) R2511.
[70]
Y. Kuramoto and T. Tsuzuki, `` Persistent propagation
of concentration waves in dissipative media far from thermal equilibrium'',
Progr. Theor. Physics
55
(1976) 365;
G.I. Sivashinsky, ``Nonlinear analysis of hydrodynamical
instability in laminar flames - I. Derivation of basic equations'',
Acta
Astr.
(1977) 1177.
[71]
L.A. Bunimovich and Ya.G. Sinai, ``Space-time chaos
in coupled map lattices'',
'Nonlinearity"
, 491 (1988).
[72]
L.A. Bunimovich and Ya.G. Sinai, ``Statistical Mechanics
of Coupled Map Lattices'', pp. 167-187 in K.Kaneko, ed.,
Coupled Map Lattices
(J.Wiley, New York 1993).
[73]
C. Boldrighini, L.A. Bunimovich, G. Cosimi, S. Frigio
and A. Pellegrinotti, ``Ising-type and other transitions in one-dimensional
Coupled Map Lattices with sign symmetry'',
J. Stat. Phys.
102
1271 (2001).
[74]
M Simoni, G. S. Cymbalyuk, M. Q. Sorensen, R. L. Calabrese,
and S. P. DeWeerth. Development of Hybrid Systems: Interfacing a Silicon Neuron
to a Leech Heart Interneuron. Advances in Neural Information Processing (2000)
13:173-179.
[75]
C. G. Wilson, R. J. Butera, C. A. DelNegro, J. Rinzel,
and J. C. Smith. Ïnterfacing computer models with real neurons: respiratory
`cyberneurons' created with the dynamic clamp." Frontiers in Modeling and
Control of Breathing: Integration at Molecular, Cellular, and Systems Levels,
C.-S. Poon and H. Kazemi, eds. Kluwer/Plenum Publishers, Boston, MA, 2001,
pages 119-124.
[76]
T.G. Brown, ``The intrinsic factors in the act of progression
in the mammal," Proc. Roy. Soc. B, London (1911) 84: 308-319.
[77]
C.C. Chow and N. Kopell, Dynamics of spiking neurons
with electrical coupling, Neural Comp. (2000) 12:1643-1678.
[78]
van Vreeswijk, C.A., Abbott, L.F., and Ermentrout,
G.B. When inhibition, not excitation, synchronizes coupled neurons. J. Computational
Neuroscience. (1995) 1:303-313.
[79]
R. C. Elson, A. I. Selvereston, R. Huerta, N. F. Rulkov,
M. I. Rabinovich, and H. D. I. Abarbanel, "Synchronous Behavior of Two Coupled
biological Neurons," Physical Review Letters (1998) 81:5692-7.
[80]
R. J. Butera, C. G. Wilson, C. DelNegro, and J. C.
Smith. Ä methodology for achieving high-speed rates of artificial conductance
injection in electrically excitable biological cells." IEEE Trans. on Biomed.
Eng. (2001) 48:1460-1470.
[81]
Potter, S. M. (2001). Distributed processing in cultured
neuronal networks. Progress In Brain Research: Advances in Neural Population
Coding. M. A. L. Nicolelis. Amsterdam, Elsevier. 130: 49-62.
[82]
Potter, S. M., A. N. Mart and J. Pine (1997). "High-speed
CCD movie camera with random pixel selection, for neurobiology research."
SPIE Proceedings 2869: 243-253.
[83]
Potter, S. M. and T. B. DeMarse (2001). Ä new
approach to neural cell culture for long-term studies." J. Neurosci. Methods
110: 17-24.
[84]
DeMarse, T. B., D. A. Wagenaar, A. W. Blau and S. M.
Potter (2001). "The Neurally Controlled Animat: Biological Brains Acting
with Simulated Bodies." Autonomous Robots 11: 305-310.
[85]
M. Dhamala, G. Pagnoni, K. Wiesenfeld, and G. S. Berns,
``Dimensional estimates from fMRI signals of the human brain vary with mental
load,'' Phys. Rev. E (accepted, 2002).
[86]
G.S. Berns, A. W. Song, and H. Mao, ``Continuous functional
magnetic resonance imaging reveals dynamic nonlinearities in dose-response
curves for finger opposition,''
J. Neurosci.
19
, RC17:1-6 (1999).
[87]
see,
e.g.
, ``The constructive role of noise
in fluctuation driven transport and stochastic resonance", special focus
issue of Chaos (volume 8), edited by R.D. Astumian and F. Moss (1998).
[88]
R.F. Fox, ``Rectified Brownian movement in molecular
and cell biology", Phys. Rev. E
57
, 2177-2203 (1998); R.F. Fox and
M.H. Choi, ``Rectified Brownian motion and kinesin motion along microtubules",
Phys. Rev. E
63
, 051901, 1-11 (2001).
[89]
F. Jaramillo and K. Wiesenfeld, ``Mechanoelectrical
transduction assisted by Brownian motion: a role for noise in the auditory
system", Nature Neuroscience
, 384-388 (1998).
[90]
K. Wiesenfeld and F. Moss, ``Stochastic resonance and
the benefits of noise: from ice ages to crayfish and SQUIDs,''
Nature
373
, 33 (1995).
[91]
M.W. Slutzky, P. Cvitanovi\'c and D.J. Mogul, ``Deterministic
chaos and noise in three in vitro hippocampal models of epilepsy,''
Annals
of Biomedical Engineering
29
, 1 (2001).
[92]
www.cns.gatech.edu/research
[93]
Georgia Tech Center for Education In Science Mathematics
and Computing,
www.ceismc.gatech.edu
[94]
Center for Computational Molecular Science & Technology,
www.ccmst.gatech.edu
[95]
www.cns.gatech.edu/centers/sisters.html
[96]
Center for Dynamical Systems & Nonlinear Studies,
www.math.gatech.edu/cdsns
[97]
www.cns.gatech.edu/centers/sorelle.html
[98]
www.cns.gatech.edu/CNS/Bylaws.html
[99]
www.math.gatech.edu/
pelesko/acelab.html
[100]
Ghrist, R., P. Holmes, and M. Sullivan, ``Knots and
Links in Three-Dimensional Flows,''
Springer Lecture Notes in Mathematics
1654
(Springer-Verlag, New York, 1997).
[101]
Ghrist, R., J. Van den Berg, and R. Vandervorst, ``Closed
characteristics of fourth-order twist systems via braids,''
C. R. Acad.
Sci. I
331
, 861-865 (2000).
[102]
L.A. Bunimovich and J. Rehacek, ``How High-Dimensional
Stadia Look like?,"
Comm. Math. Phys.
, 197 (1998).
[103]
Ghrist R. and D. Koditschek, ``Safe, cooperative robot
dynamics on graphs,'' to appear in
SIAM J. Cont. & Opt.
, ArXiv
preprint:
cs.RO/0002014
(2001).
[104]
V. I. Arnol'd, V. V. Kozlov, and A. I. Neishtadt,
Mathematical
Aspects of Classical and Celestial Mechanics
(Springer-Verlag, New York,
1988).
[105]
L. A. Bunimovich L.A., ``Many-dimensional nowhere dispersing
billiards with chaotic behavior,''
Physica D
33
, 58 (1988).
[106]
L. A. Bunimovich, ``On absolutely focusing mirrors,
In: 'Ergodic Theory and Related Topics,'' (ed. by U. Krengel
et al
),
Lecture Notes Math.
1514
, 62 (1992).
[107]
P. Grosfils, J.-P. Boon, E. G. D. Cohen, L. A. Bunimovich,
``Propagation and self-organization in lattice random media,"
J. Stat. Phys.
97
, 575 (1999).
[108]
L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua,
``Methods of Qualitative Theory in Nonlinear Dynamics," Part II, (World Sci.,
London, 2001).
[109]
L.A. Bunimovich, ``Lattice Dynamical Systems," in
From
Finite to Infinite Dynamical Systems,
J.C. Robinson and P.A. Glendinning,
eds., p. 59 (Kluwer, London 2001).
[110]
P. Bak, C. Tang, and K. Wiesenfeld, ``Self organized
criticality,''
Phys. Rev. A
38
, 364-374 (1988).
[111]
T. Heath, K. Wiesenfeld and R. York, ``Manipulated
synchronization: beam steering in phased arrays,''
Int. J. Bif. Chaos
10
, 2619-2627 (2000).
[112]
K. Wiesenfeld, P. Colet, and S. Strogatz, ``Frequency
locking in Josephson arrays: connection with the Kuramoto model,''
Phys.
Rev. E
57
, 1563-1569 (1998).
[113]
J. L. Rogers, M. F. Schatz, O. Brausch, and W. Pesch,
``Superlattice patterns in vertically oscillated Rayleigh-Bénard convection,''
Phys. Rev. Lett.
85
, 4281 (2000).
[114]
R. O. Grigoriev, ``Symmetry and control: spatially
extended chaotic systems,''
Physica D
140
, 171-192 (2000).
[115]
B. Bamieh, F. Paganini, and M. Dahleh ``Optimal control
of distributed arrays with spatial invariance,''
Lect. Notes Contr. Inf.
245
, 329-343 (1999).
[116]
R. O. Grigoriev, M. C. Cross, and H. G. Schuster, ``Pinning
control of spatiotemporal chaos,''
Phys. Rev. Lett.
79
, 2795-2798
(1997).
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