This talk will address a number of experimental
observations or measurements pertaining to pattern formation in Rayleigh-B\'enard
convection (RBC) which fall into one or both of the following categories:

1.) They seem interesting but have never
been addressed theoretically

2.) They disagree with theoretical predictions.

Included are:

a.) the re-emergence of relatively simple
patterns in cylindrical samples at large Rayleigh numbers

b.) Unorthodox correlation-length scaling
and defect formation in domain chaos, i.e. in RBC with rotation

c.) Formation of "squares" where K\"uppers-Lortz-unstable
rolls are predicted in RBC with rotation

d.) A subcritical bifurcation without hysteresis
in RBC with rotation

e.) Absence of convection above a predicted
Hopf bifurcation in RBC with rotation

Granular materials present a host of challenging
questions at the most basic level. In dense granular materials, complex
force structures, known as force chains dominate the transmission of force.
Competing models to describe force propagation include purely diffusive,
elastic, and wave-like behavior. One way to address the issue of how stresses
propagate is to carry out experimental determinations of the Green's function,
i.e. the response of a material to a local force perturbation. We have
recently carried out such measurements using 2D photoelastic particles
that allow us to determine the local force at the particle scale. These
measurements indicate that the ensemble-averaged response depends significantly
of the amount of order in the packing. In highly ordered packings, the
response is consistent with a wave-like propagation of forces, whereas
in disordered packings, the response is elastic. Notably, any given realization
is typically complex, with large deviations from the ensemble-averaged
response. When dense materials are deformed, the stress chains break and
reform, leading to large-scale fluctuations. In Couette shear experiments,
we have found that there is a novel transition with second-order-like properties
as the density of the sample is varied. Conventional Coulomb models for
stresses in shearing materials indicate that the forces should be independent
of shear rate. However, we have recently found a slow dependence on shear
rate that is not accounted for by conventional models.

Mathematical bounds on mass, momentum and
heat transport for solutions of the incompressible Navier-Stokes and Boussinesq
equations are reviewed. In some cases the scaling indicated by the bounds
is (nearly) saturated by turbulent flows, but in some cases---most notably
turbulent Rayleigh-Benard convection---there remains a large gap between
experiments and the rigorous heat transport limits. We discuss some current
research exploring these issues further.

Turbidity currents are the primary agents
of coarse-grained sediment transport across the continental slope and abyss.
These are gravity currents which flow when sediment suspended in water
is more dense than the ambient fluid (usually sea or lake water). As such,
their behavior is very sensitive to changes in mass and momentum, which
are functions of sediment concentration, grain-size distribution, current
height, and gradient. The critical threshold of erosion is called "ignition",
and produces rapid acceleration and bulking up to a maximum 10% sediment
concentration at which point the primary suspension mechanism, turbulence,
is damped. Below the ignition threshold, turbidity currents will decelerate
and dampen to predicable termination. These strong positive and negative
feedbacks can be indirectly measured in geological and experimental systems
and the system dynamics qualitatively characterized. The relevant parameters
can be collapsed to produce simple phase diagrams that help predict grain-size
distributions, bed characteristics, pore volume connectivity, and behavioral
response. Numerical, experimental, theoretical, and field-based studies
will help to place quantitative constraints on system thresholds and response.

This will be a general introduction, with
emphasis on biomedical applications.

After an introduction where the two paradigms
of stochasticity and deterministic chaos are opposed, and several physiological
examples are sketched, I'll start the more technical part by dicsussing
time delay embeddings and choices of parameters for them.

A first application will deal with noise
reduction and signal separation based on the geometry of embeddings. Fetal
heart beat extraction from a univariant ECG signal is discussed as a special
case.

We then discuss classical invariants (metric
entropy, attractor dimension, Lyapunov exponents) and argue why using them
as indicators for chaotic determinism is not very useful. The same should
be true also for alternatives like false nearest neighbors or forecasting
errors. In contrast we shall argue that strict determinism is not needed
for the arsenal of nonlinear time series analysis to be useful. In contrast,
I shall present evidence that effective "attractor" dimensions can be useful
for predicting epileptic seizures and localizing epileptic foci.

Finally we shall discuss various methods
to study interdependencies between different time series. This includes
cross correlation and coherence, mutual information, phase synchronization,
and other interdependence measures. We shall discuss their usefulness in
EEG analysis, in particular for epilepsy patients. Among these measures,
of particular interest are asymmetric measures because they could, independent
of time delays, indicate causal connections. Again this is illustrated
with epileptic EEGs.

Recent experimental and theoretical work
on the branching of electron flow under the influence of soft potential
energy hills and valleys in nanostructures will be presented (Nature, March
8 2001). The work is applicable to a variety of situations involving propagation
through random media, and represents perhaps a new regime in chaos theory.
The cumulative effect of long range travel over many correlation lengths
of the potential surprisingly leaves strong, preferred branches of flow
intact. The combination of such flow with billiard walls will be discussed.
Theoretical foundations and quantum implications will be presented.

The promise of computational biology is
that it provides an enhanced ability to relate changes at the molecular
level to changes in macroscopic function. But as with all techniques, computational
modeling and simulation involve making significant tradeoffs, usually
compromising realism to maintain tractability. In this talk, I will discuss
the challenges of both creating and analyzing large scale, biologically
realistic models of atrial fibrillation and explore some of the reasons
why simulation and experimentation must come closer together to fully elucidate
the dynamics of wavefront conduction in inhomogeneous, three-dimensional
domains.

We study the motion of fluids, with the
aim of developing a fundamental understanding of fluid flow. Our program
is characterized by close cooperation among experimenters, theoreticians,
and simulators. The world about us exhibits many beautiful and important
fluid flows. Consider clouds and waves, storms, and earthquakes, sunspots
and mountain-building. What can we learn from all this richness?

Mostly our work involves solving particular
problems, e.g. 'how does heat flow in a pot of water heated over a flame'.
But, in following these problems we soon get to broader issues: predictability
and chaos, the likelihood of very extreme outcomes, and the natural formation
of complex 'machines'.

In the end, we try to ask if there is a
'science of complexity' and are there natural 'laws' of complex things.
My answer is 'no', but I do see important lessons to be learned from studying
such systems.

In the past several decades, physicists
have made great strides in understanding how spatial patterns can arise
in systems driven far from equilibrium. Of course, many important issues
and significant challenges remain. But, with this sense of progress, many
researchers began addressing the question of whether the study of pattern
formation could help elucidate the formation of structure in biological
systems, often called morphogenesis. Of course, living matter is much more
complex than non-living. Yet, this talk will hopefully convince you that
not only is this physics-based approach possible, but is in fact extremely
promising.

There are many processes one could choose
to discuss; for definiteness, I will focus on the life cycle of the soil
amoeba Dictyostelium discoideum. In this organism, starvation triggers
a day-long series of transformations that take solitary amoebae and create
a cooperative multicellular organism; the process culminates in a plant-like
fruiting body containing spore cells specialized for survival in harsh
conditions. Ideas from the physics of pattern formation have been used
to help explain the wave field used for cell guidance, the streaming of
cells into the aggregate and the collective motions seen in multicellular
stages. Currently, several groups are working on the single-cell chemotactic
response from a similar perspective.

The
understanding of the dynamics of turbulent flows has been a major goal
for fundamental and applied fluid dynamics research for almost a century
now. On the fundamental side, turbulence is the head figure of a non-linear
dissipative system with a very large number of degrees of freedom. On the
applied side, the properties of turbulent flows govern the dispersion of
pollutants, the physics of mixing, etc. In very recent years, analytical
and numerical studies have shown that progress can be made by analyzing
the flow properties in the reference frame of a moving fluid particle (the
Lagrangian viewpoint), instead of considering the velocity field at a fixed
point in space (the Eulerian viewpoint).

In
order to completely address turbulence in the Lagrangian frame, one needs
to describe the dynamics over the entire range of scales of motion. We
have developed such technique, based on sonar principles, to measure directly
the velocity of individual small tracer particles over long times. We have
analyzed the statistics of the Lagrangian velocity of single particles
for flows with turbulent Reynolds numbers between 100 and 1100. We observe
that the Lagrangian spectrum has a Lorentzian form in agreement with a
Kolmogorov-like scaling in the inertial range. The probability density
function (PDF) of the velocity time increments displays a change of shape
from quasi-Gaussian a integral time scale to stretched exponential tails
at the smallest time increments. This intermittency, when measured from
relative scaling exponents of structure functions, is more pronounced than
in the Eulerian framework.

Another
important observation is that in the erratic course of the particle motion,
infinitesimal changes of velocity occur with `random' decorrelated directions
but with a correlation of magnitude which persists over the longest times
of the flow. Using an analogy with the properties of Multifractal Random
Walks, we propose that this feature is essential in the development of
intermittency in turbulence.

We study quantum dynamics of ultra-cold
cesium atoms in mixed phase space consisting of islands of stability surrounded
by chaos. We use a new method to prepare a minimum uncertainty wavepacket
located on one island in phase space. We observe coherent tunneling oscillations
between this state and a symmetry related island. We show that this system
exhibits chaos assisted tunneling as characterized by the participation
of the intermediate stochastic sea, the sensitivity to parameters, and
the enormous enhancement in the tunneling rate between distant states.

In three-dimensional (3D) turbulent flow,
vortices stretch axially and fold, but this process cannot occur in two
dimensions. While all turbulent flows are 3D on sufficiently small scales,
atmospheric and oceanic flows are approximately 2D on large scales. We
study turbulence in a rotating tank where the flow becomes 2D for sufficiently
rapid rotation rate (by the Taylor-Proudman theorem), while for low rotation
rates the flow is 3D [1]. We find that for 2D turbulence the probability
distribution function (PDF) for the difference in velocity between two
points is independent of the separation r between the two points, i.e.,
the flow is self-similar. In contrast, the PDFs for 3D turbulence are gaussian
for large r and exponential for small r. We further compare the 2D and
3D turbulence flows by determining structure function scaling exponents
and by applying the beta and gamma tests of the hierarchical structure
model; these quantities will be defined and discussed. The conclusion is
that 2D turbulence in a rotating flow is surprisingly intermittent, but
the intermittency is a consequence of large coherent vortices rather than
the stretching and folding of vortex lines as in 3D.

*Supported by ONR

[1] C.N. Baroud, B.P. Plapp, Z.S. She,
and H. L. Swinney, submitted

Recent interest in microscale flows, which
can sustain exceedingly large surface to volume ratios, has focused attention
on the use of normal or shear force actuation to effect liquid migration
along a solid surface. For instance, a thin liquid film supported on a
differentially heated substrate will spontaneously flow toward the cooler
end in a process known as thermocapillary forcing. In another example,
a film contacted by a non-uniform distribution of surface active material
will rapidly flow toward the uncontaminated end under the action of Marangoni
stresses. Such spreading films typically undergo fingering instabilities
at the advancing front which resemble either a series of parallel liquid
rivulets or highly ramified dendritic structures. During the past several
years, we have used a combination of experiment and theoretical modeling
in an effort to provide a unified framework describing the stability characteristics
of these and related thin film flows. The presence of capillary, thermocapillary,
Marangoni or van der Waals stresses in free surface films creates spatially
(and temporally) dependent interface shapes. The linearized operators governing
disturbances in film thickness or concentration are therefore typically
non self-adjoint. A consequence of this feature is that conventional modal
analysis can only describe the asymptotic behavior of these

systems. A rigorous description of the
early and intermediate behavior requires a transient growth study. This
type of analysis not only identifies the growth rate of optimal disturbances
but reveals their initial and evolved waveform shape at all times. Strongly
non-normal operators can introduce the possibility of severe disturbance
amplification and subsequent non-linear coupling.

In this presentation, we outline the transient
growth behavior and amplification of optimal disturbances for thermocapillary
driven flows. Time permitting, we review Marangoni driven systems as well.
We examine the pseudospectral behavior of the associated disturbance operators
and illustrate why transient growth studies offer a more suitable probe
of the stability of free surface thin film flows.

The myth `bumble-bees can not fly according
to conventional aerodynamics' simply reflects our poor understanding of
unsteady viscous fluid dynamics. In particular, we lack a theory of vorticity
shedding due to dynamic boundaries at the intermediate Reynolds numbers
relevant to insect flight, typically between $10^2$ and $10^4$, where both
viscous and inertial effects are important. In our study, we compute unsteady
viscous flows, governed by the Navier-Stokes equation, about a two dimensional
flapping wing which mimics the motion of an insect wing. I will present
two main results: the existence of a preferred frequency in forward flight
and its physical origin, and 2) the vortex dynamics and forces in hovering
dragonfly flight. If time permits, I will show the recent results on comparing
our computational results against robotic fruitfly experiments and modeling
three dimensional flapping flight driven by muscles.

The turbulence of planetary atmospheres
and oceans self-organizes into a spatio-temporal pattern of coherent structures
such as vortices and jets. These structures provide insight into the long-standing
problem of reducing fluid turbulence to a chaotic dynamical system. The
attractor of the turbulence is an evolving population of structures, and
the structures' degrees of freedom are the reduced coordinate system which
describe the attractor. These ideas are explored in a hierarchy of systems
with increasing complexity, from Hamiltonian ordinary differential equations
to oceanic observations.

We consider chaotic dynamics of tracers
in a system of point vortices and, in parallel, the phase space topology
of the vortices. There exist a specific connection between coherent structures
of the vortex system and tracers transport , which is possible to describe
in an analytical way and confirm by simulations. The coherent structures
occur as clusters of few vortices , which impose the tracers kinetics of
the fractional type. On that way a characteristic exponent of the tracers
dispersion can be obtained from the first principles.