% project.tex                           NSF Appl Math proposal

% Predrag (Marsha's edits)              Nov  4 2003
% Predrag                               Nov  2 2003
% Marsha                                Oct 28 2003
% Predrag                               Oct 26 2003
% based on project.tex NSF Math Physics Sep 23 2003
% derived from
% MURI03 proposal
% PFCNSproject.tex and McDonnell Fundation proposal
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\noindent
{\bf \large  Project description:

\medskip

\noindent
\Large  \Title
}

\medskip

\noindent
{\bf \large  Predrag Cvitanovi\'c, School of
Physics, Georgia Institute of Technology }

\medskip

% \section{Introduction}
% \label{s:chaos}

Pushing the current limits of what
is numerically attainable in solving large sets of PDEs,
the best we  currently can do is to
write down a set of coupled hydrodynamical/chemical
PDEs, discretize it on a mesh or a basis function set of order of
$10^4$-$10^9$
components,  with velocity and concentration fields
at each point,
and follow numerically trajectories in high-dimensional
phase spaces.
Due to the sensitive dependence on initial conditions, and
dramatic qualitative differences in dynamics
at different model parameter values,
already at moderate Rayleigh numbers
the available nonlinear dynamics tools largely fail.

Nevertheless, insights gleaned from nonlinear dynamics
have greatly changed the ways we think about turbulence.
The field theories that describe the classical world ---
the strongly nonlinear
hydrodynamics,
magneto-hydrodynamics,
% Burgers,
complex Ginzburg-Landau equations
---
are turbulent
and have bewildering wealth of solutions, with very few of the important
ones analytical in form.


A systematic exploration of solutions of such systems has so far
been implemented only  for one of the very simplest field theories,
the 1-dimensional Kuramoto-Sivashinsky system,
% from ks.tex, PC            30/6/96
within what we shall here refer to as the ``Hopf's vision'' or ``recurrent patterns program''.
In its spirit, this program is very different from most ideas that animate
current turbulence research.
It is distinct from
the Landau quasi-periodic picture of turbulence as a sum of
infinite number of incommensurate frequencies.
% with dynamics taking place on a large-dimensional torus.
It is not the
Kolmogorov's 1941 homogeneous turbulence, with no
coherent structures fixing the length scale, here all the action is
in specific coherent structures.
It is emphatically {\em not} universal; spatiotemporally periodic solutions
are specific to the particular set of equations and boundary conditions.
And it is {\em not} probabilistic; everything is fixed by the deterministic
dynamics, with no probabilistic assumptions on the velocity distributions
or external stochastic forcing.


The theory of deterministic chaos
tells us that the starting approximation to strongly
non-linear systems should be quite different from the traditional
point of departure, the nearly periodic motions. Portraits of
chaotic systems exhibit amazingly rich self-similar
structure which is not at all apparent in their
formulation in terms of differential equations.
The theory proposed here follows a path radically different from
direct long-time PDE simulations.
It postulates, in a nutshell, that
 $(\mbox{mesh size} \times \mbox{time})^{\mbox{\# fields}}
     \approx 10^{10}$
 is a vastly too large a number, and that a finite set of {\em building
blocks}, of order of $10^2-10^4$ recurrent patterns, is all that is {\em
required for the eventual assembly of a predictive capability for
non-equilibrium turbulent flows}.

A method for numerical determination of recurrent patterns
is described in \refsect{s:RecurFind}.
In \refsect{s:KurSivSyst} we introduce the
Kuramoto-Sivashinsky system, and in
\refsect{s:planeCouetteTurb} we review the status of determining
such patterns for full  three-dimensional Navier-Stokes
turbulence.
The theory of computing averages in spatio-temporally chaotic dynamics
in terms of spatio-temporally recurrent unstable
patterns is reviewed in \refsect{s:PerOrsInAction}, and the detailed
plan of the proposed research is outlined in \refsect{s:PropRes}.


\section{Background}

The program that we propose to follow here was lucidly outlined by
% German mathematician
Eberhardt Hopf in a beautiful 1948 paper~\cite{Hopf48}; his
prose is as good as anything written since on the subject, so we
introduce the program in Hopf's own words:
\PC{fix 1951 to 1947?}

{\em ``Consider an incompressible and homogeneous viscous fluid within
given material boundaries under given exterior forces. The boundary
conditions and the outside forces are assumed to be stationary, i.e.
independent of time. For that, it is not necessary that walls be at rest
themselves. Part of the material walls may move in a stationary movement,
provided that the geometrical boundary as a whole stays at rest. An
instance is [$\cdots$]  a fluid between two parallel planes which are
translated within themselves with given constant velocities.''
    }% end {\em {quote}

In \refsect{s:planeCouetteTurb} we  review the current status of this
problem, ``the constrained plane Couette turbulence'', and explain
why we believe that today Hopf's program can be implemented.

% ($\cdots$)

E.~Hopf continues: {\em ``It is convenient to visualize the solutions in
the phase space $\Omega$ of the problem. A phase or state of the fluid is
a vector field $u(x)$ in the fluid space that satisfies [the Navier-Stokes
equations] and the boundary conditions. The totality $\Omega$ of these
phases is therefore a functional space with infinitely many dimensions. A
flow of the fluid represents a point motion in $\Omega$ and the totality
of these phase motions forms a stationary flow in the phase space
$\Omega$, which, of course, is to be distinguished from the fluid flow
itself. What is the asymptotic future behavior of the solutions, how does
the phase flow behave for $ t \to \infty$~? And how does this behavior
change as [the viscosity] $\mu$ decreases more and more? How do the
solutions which represent the observed turbulent motions fit into the
phase picture? The great mathematical difficulties of these important
problems are well known and at present the way to a successful attack on
them seems hopelessly barred. There is no doubt, however, that many
characteristic features of the hydrodynamical phase flow occur in a much
larger class of similar problems governed by non-linear space-time
systems. In order to gain insight into the nature of hydrodynamical phase
flows we are, at present, forced to find and to treat simplified examples
within that class.''
    }% end {\em {quote}


One of the simplest and today extensively studied spatially extended
dynamical systems, reviewed here in \refsect{s:KurSivSyst}, is the
1-spatial dimension Kuramoto-Sivashinsky\rf{KurSiv} system
that describes the flutter of a gas flame.
For this
relatively simple system, we have checked Hopf's vision in
detail~\cite{CCP96}: many recurrent patterns have been determined
numerically, and the recurrent-patterns theory predictions tested for
various values of flame front parameters. But, as we shall argue in
\refsect{s:planeCouetteTurb}, with modern computation and new ideas, the
way to a successful attack on the full Navier-Stokes problem is no longer
``hopelessly barred''.


E.~Hopf: {\em ``The observational facts about hydrodynamic flow reduced to
the case of fixed side conditions and with $\mu$ as the only variable
parameter are essentially these: For $\mu$ sufficiently large, $\mu >
\mu_0$, the only flow observed in the long run is a stationary one
(laminar flow). This flow is stable against arbitrary initial
disturbances. Theoretically, the corresponding exact solution is known to
exist for every value of $\mu > 0$ and its stability in the large can be
rigorously proved, though only for sufficiently large values of $\mu$. The
corresponding phase flow in phase space $\Omega$ thus possesses an
extremely simple picture. The laminar solution represents a single point
in $\Omega$ invariant under the phase flow. For $\mu > \mu_0$, every phase
motion tends, as $ t \to \infty$~, toward this laminar point. For
sufficiently small values of $\mu$, however, the laminar solution is never
observed. The turbulent flow observed instead displays a complicated
pattern of apparently irregularly moving ``eddies'' of varying sizes. The
view widely held at present is that, for $\mu > 0$ having a fixed value,
there is a ``smallest size'' of eddies present in the fluid depending on
$\mu$ and tending to zero as $\mu \to 0$. Thus, macroscopically, the flow
has the appearance of an intricate chance movement whereas, if observed
with sufficient magnifying power, the regularity of the flow would be
never doubted.

The qualitative mathematical picture which [E.~Hopf] % author
 conjectures to correspond to the known facts about hydrodynamic flow is
this: To the flows observed in the long run after the influence of the
initial conditions has died down there correspond certain solutions of the
Navier-Stokes equations. These solutions constitute a certain manifold
 \PC{LaTeX for gothic $M$?}
 $\inertM = \inertM(\mu)$ invariant under phase flow. Probably owning to
viscosity $\inertM$ has a finite number $N= N(\mu)$ of dimensions. This
effect of viscosity is most evident in the simplest case of $\mu$
sufficiently large. In this case $\inertM$ is simply a point, $N=0$. Also
the complete stability of $\inertM$ is in this simplest case is obviously
due to viscosity. On the other hand, for smaller and smaller values of
$\mu$, the increasing chance character of the observed flow suggests that
$N \to \infty$ monotonically as $\mu \to 0$.''
    }% end {\em {quote}

Today $\inertM(\mu)$ is known as the ``inertial manifold'' and widely
studied. Its finite dimensionality for non-vanishing $\mu$ has been,
in certain settings,
rigorously established by Foias and collaborators~\cite{Foias88}.
Hopf's ``monotonicity'' with system parameters such as $\mu$
is a dream - a family of dynamical systems can do anything, including collapsing to a
1-dimensional periodic attractor, infinitely often~\cite{FST86}.
Nevertheless, the intuition is qualitatively correct.
Inspired by his study of the first (Hopf) bifurcation from stationary
to oscillatory behavior, E.~Hopf
proposes in the 1948 paper what is today known as
the Landau-Hopf model of turbulence~\cite{Hopf48,landau44},
with turbulence arising from an infinite sequence of Hopf bifurcations
of incommensurate frequencies, and the dynamics taking place on a
large-dimensional torus. In hindsight, this is not a good model.
In a fundamental 1971
paper Ruelle and Takens~\cite{rue_tak} have shown that
nonlinearities lead to turbulent
dynamics confined to a strange attractor $\inertM(\mu)$  of fractal dimension.
Our claim is that the hierarchy of recurrent patterns of longer and longer
periods cover this fractal set in a systematic fashion.

% Wrong:
% Discusses how the dimension of $\inertM$ grows
% as through bifurcations new manifolds $\inertM(\mu)$
% are continuously branching away.

What E.~Hopf writes next is prescient:
%
{\em ``The geometrical picture of the phase flow is, however, not
the most important problem of the theory of turbulence. Of greater
importance is the determination of the probability distributions
associated with the phase flow [$\cdots$].''
%  , particularly of their asymptotic limiting
%  forms for small $\mu$.
    }% end {\em {quote}
This is the key challenge, the one in which the theory of dynamical
systems has made the greatest progress in the last half century:
the Sinai-Ruelle-Bowen
ergodic theory of ``natural'' or SRB measures for
far-from-equilibrium systems.
This is the challenge the proposal addresses
employing the methodology sketched in
\refsect{s:PerOrsInAction}, and developed in depth
in {\tt ChaosBook.org}, our electronic textbook.~\cite{QCcourse}
\PC{insert loop.tex}
% \begin{quote}
% ``
% ($\cdots$)
% ''
% \end{quote}

\section{Recurrent patterns: a repertoire of turbulence?}

 %
\begin{wrapfigure}{r}{2.4in}
\vspace{-3mm}
\includegraphics[width=2.4in]{figs/FigVortex.eps}
\caption{\small\em
  Planktonic copepods accelerate their swimming appendages up to $50
  m/s^2$. Two vortices are created by 4 pairs of legs
  galloping twice within the field of view~\cite{JY02}.
\vspace{-3mm}
}
\label{Fig-JY_1vortex}
\end{wrapfigure}
 %
Spatially extended systems are systems with very many coupled degrees
of freedom whose dynamics range from ordered to very disordered and turbulent.
Theoretical challenges in learning to control such systems
require understanding the instability mechanisms arising
from the interplay between nonlinearity, transient dynamics, and stochasticity, often resulting in rather surprising dynamics. For instance, the onset of turbulence observed in shear flows occurs well within the Reynolds number
regime where the laminar flow is still linearly stable. Transient
amplification in such systems leads to extreme sensitivity to
noise,
%increasing exponentially with the size of the system and
precluding any spatially localized control, even at physically
unrealistic low levels of noise~\cite{evap}.

The inspiration for the ``recurrent patterns program''
comes from the way we perceive turbulence. Turbulent
systems never
 settle down, but we can identify a snapshot as a ``cloud'', and an
experimentalist can tell what the values of physical parameters in a
turbulence experiment were after a glance at the digitized image of its
output.  Still more intriguingly, even in a turbulent sea, a plankton can
identify a prey, predator, or mate by the complex but characteristic wakes
that bodies moving through fluid shed, \reffig{Fig-JY_1vortex}.
 How do we do it?

An answer was offered by E. Hopf~\cite{HopfsLastSrc}.
% (so says E.A.~Spiegel~\cite{MS66,EAS87}, who has pioneered study
% of turbulence in terms of unstable periodic solutions
%  -- I admit to not having been able to track down {\em where}
% Hopf writes this).
In Hopf's vision turbulence explores a
repertoire of distinguishable patterns; as we watch
a turbulent system evolve,
every so often we catch a glimpse of a familiar whorl,
\reffig{Fig-recurrent}.

At any instant and a given finite spatial resolution the system
approximately tracks for a finite time a pattern belonging to a finite
alphabet of admissible patterns, and the dynamics can be thought of as a
walk through the space of such patterns, just as chaotic dynamics with a
low dimensional attractor can be thought of as a succession of nearly
periodic (but unstable) motions.
%
There are two issues here:
%
%\noindent
%\parbox[t]{4.00in}
%{\noindent
 (1) {\em how} to identify the alphabet of patterns?
%\\
 (2) {\em what} to do with this repertoire of patterns?
%}


\section{Recurrent patterns: how to find them?}
\label{s:RecurFind}

 %
\begin{wrapfigure}{r}{3.0in}
\vspace{-3mm}
\hspace{-1ex}
\includegraphics[width=0.23\textwidth]{figs/kscc1.eps}~\includegraphics[width=0.23\textwidth]{figs/kscc2.eps}
\hspace{-4ex}
\vspace{-4mm}
\caption{\small\em
 A pattern and its near recurrence in a Kuramoto-Sivashinsky simulation:
the swirl on the left recurs approximately within another pattern,
right image. The color/greyscale codes the height of the flame front.~\cite{LanDescent}
    }
\vspace{-3mm}
\label{Fig-recurrent}
\end{wrapfigure}
%
The issue of {\em ``how''} is an intensely numerical
undertaking. Consider what is at stake:
We need to compute,  or extract from data, many
distinct 3-dimensional color videos of patches of
turbulence.
Just in terms of memory we need something of order of
%$(\mbox{length}^3 \times \mbox{time})^{\mbox{d.o.f} \times \mbox{color}}=$~
a gigabyte to store a single pattern.
% (Here d.o.f. = degrees of freedom
% at each point: velocities, reactant concentrations, temperature, etc.).
If a given image is an instant in
the motion of a turbulent swirl, nature computes the next instant
by letting every particle jostle its neighbors. We do it by
time consuming numerical pixel-by-pixel calculations. That is still not so hard,
but the theory proposed here
seeks a 3-dimensional pattern that
repeats itself {\em exactly} after a (yet to be determined) time period.
{\em A priori},
we have no clue what such
patterns look like, how big and what shapes are the basic spatial units,
or what their time periods should be.
So in our initial guess image
{\em all}
 pixels
are wrong, and we need to keep jostling them numerically until
gigabytes of pixels settle into a pattern allowed by
the laws of motion. Not only that, but a computer needs variable
resolution, as small regions can play a very important role.
All that takes
computer time, lots of it. The theory then requires that we
compute many such patterns, and be sure that we have found
{\em all} of the patterns up to given spatial extent and time period
within the desired tolerance limits.

Clearly, conceptually new methods
applicable to high numbers of
deterministic degrees of freedom as well as to mixed systems of
chaotic, integrable, and stochastic components need to be developed;
the present dearth of
such methods remains a fundamental barrier to the direct application
of concepts of nonlinear science to turbulent phenomena.


\subsection{Recurrent pattern searches}


We make here a distinction between a {\em periodic orbit}
--- a low-dimensional trajectory that folds
into itself after a time $\period{}$,
and a {\em recurrent pattern} --- a trajectory in the infinite-dimensional
state space $\pS$
which tiles both space and time periodically.
A periodic orbit is a solution $(\pSpace,\period{})$
of the {\em  periodic orbit condition}
$
f^{\period{}}(\pSpace) = \pSpace
$
for a given flow $\pSpace \to f^{t}(\pSpace)$.
The task we face is to determine a set of recurrent patterns of
shortest periods. Any numerical solution of a PDE is based
on its
representation in terms of a truncated but
large set of coupled nonlinear ODEs. So, in practice
we always search for {\em periodic
orbits} of flows defined by first order ODEs
\beq
\frac{d\pSpace}{dt}=\pVeloc(\pSpace)
    \,,\qquad
 \pSpace \in \pS \subset \mathbb{R}^d
\label{p-1}
\eeq
in many (even infinitely many) dimensions $d$, with
the vector field $\pVeloc(\pSpace)$
a smooth, differentiable field.
Nevertheless, we insist on referring to the problem at
hand as a ``recurrent pattern search'', as attempts to visualize
a solution in a 15,000-dimensional
arbitrarily discretized spaces are a fruitless undertaking.
The solutions can be understood only as fields embedded
in the physical space and time.

A variety of
methods for determining all unstable periodic orbits up to a given length
have been devised and successfully implemented for low-dimensional
systems~\cite{QCcourse}.
For turbulent flows, with high
(or infinite) dimensional phase spaces and complicated dynamical behavior,
most of the existing methods become unfeasible in practice.
The bottleneck
for the recurrent patterns program
has been the lack of methods for finding even the simplest
recurrent patterns, and the lack of intuition as
to what such patterns would look like.

PI's group has formulated and explored numerically a novel
variational principle for determining unstable
spatio-temporally periodic solutions of
extended systems~\cite{crete03,LanDescent}.
The idea of the method
is to make a rough but informed guess of what the desired pattern
looks like globally, and then use a variational method to drive the
initial guess toward the exact solution. For robustness, we replace the
guess of a single  pattern at a given instant by a guess of an entire orbit. For
numerical safety, we replace the Newton-Raphson iteration  by the
``\descent'', a differential flow that strictly
minimizes a {\costFct} computed from the
deviation of the approximate flow from the true flow.

We start by guessing a
{\em loop},
 a smooth, differentiable closed curve
$\lSpace(s)\in \Loop \subset \pS$,
parametrized by $s \in [0,2\pi]$ with $\lSpace(s)=\lSpace(s+2\pi)$, and
the loop tangent vector
\[
\lVeloc(\lSpace)=\frac{d \lSpace}{ds}\,, \quad \lSpace=\lSpace(s) \in \Loop
\,.
\]
The initial loop is {\em not} a solution of the flow equations
--- it is at best a rough guess as to what a solution
might look like.

While there is no extremal principle associated with
the general flow \refeq{p-1}, at least three separate lines
of argument (cost minimization~\cite{crete03},
over-damped Newton method~\cite{LanDescent},
Wiener-Onsager-Machlup
stochastic path extremization~\cite{Onsager53})
all lead to the same variational principle, minimization of
the simplest {\costFct}
\beq
\costF[\lSpace] = {1 \over |\Loop|}
            \oint_\Loop ds\,(\lVeloc-\pVeloc)^2
    \,,\quad
    \lVeloc = \lVeloc(\lSpace(s,\tau))\,,\,\,\,
    \pVeloc = \pVeloc(\lSpace(s,\tau))
\,,
\label{loopCostFct}
\eeq
which penalizes misorientation of the local loop tangent
vector $\lVeloc(\lSpace)$
relative to the dynamical velocity field $\pVeloc(\lSpace)$
of \refeq{p-1}, see \reffig{f:loops}~(d).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t] %[h]
\centering
(a)
 \includegraphics[width=2.5cm]{figs/loop.eps}
\hspace{0.05in}
(b)
\includegraphics[width=3.3cm]{figs/porbit.eps}
(c)\,\,
\includegraphics[width=2.8cm]{figs/velocField.eps}
\hspace{0.01in}
(d)
    \includegraphics[width=3.2cm]{figs/tube.eps}
\caption{\small\em
 (a) A loop $\Loop$ {\em defines} its tangent velocity vector $\lVeloc$.
 (b) A periodic orbit $p$ is {\em defined}
    by the vector field $\pVeloc(\pSpace)$.
(c) In general the orientation of the loop
tangent $\lVeloc(\lSpace)$ does not coincide with
the orientation of the velocity field $\pVeloc(\lSpace)$;
for a periodic orbit $p$ it  does so at every $x \in p$.
(d) An annulus $\Loop(\tau)$ with the fictitious time flow $\dot{\lSpace}$
connecting smoothly the
initial loop $\Loop(0)$ to a periodic orbit $p$.
        }
\label{f:loops}
% (c) (d) were in \label{f:velocField}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Vary the  \costFct al $\costF[\lSpace]$
 with respect to the (yet undetermined) fictitious time $\tau$,
\[ %\beq
{d \costF \over d\tau} = {2 \over  |\Loop|} \oint_\Loop ds\,
    (\lVeloc-\pVeloc)\,
    {d~ \over d\tau}
    (\lVeloc-\pVeloc)
\,.
\] %\eeq
The simplest, exponentially  decreasing
 \costFct al
is obtained by taking the $\lSpace(s,\tau)$ dependence on
$\tau$ to be point-wise proportional to the deviation of the
two vector fields
\beq
\frac{d}{d \tau}(\lVeloc-\pVeloc)=
 -(\lVeloc-\pVeloc)
\,,
\label{fm0}
\eeq
so the fictitious time flow drives the
loop to $\Loop(\infty)=p$ monotonically, at exponential rate
\reffig{f:loops}~(d):
\beq
\lVeloc-\pVeloc=   e^{-\tau} (\lVeloc- \pVeloc)|_{\tau=0}
\,.
\label{fm}
\eeq

Making the $\lSpace$ dependence in \refeq{fm0} explicit
we obtain the  {\em \descent\ equation}, a PDE
which evolves the initial
loop $\Loop(0)$ into the desired periodic orbit $p$
in the fictitious time $\tau \to \infty$:
\beq
\frac{\partial^2 \lSpace}{\partial s \partial \tau}-\Mvar \frac
{\partial \lSpace}{\partial \tau}
    =
        \pVeloc-\lVeloc
\,,\qquad
 \Mvar_{ij}(x)=\frac{\partial \pVeloc_i(x)}{\partial x_j}
\,.
\label{bd5}
\eeq
Numerically, we distribute many points along a smooth loop $\Loop$
(successive snapshots of the the pattern at successive time instants).
The infinitesimal time step and loop deformation limit
corresponds to this partial differential equation. It is important to note that
we do not need to solve this equation accurately along the way --- only
the periodic orbit itself needs to be computed with a high precision.

We start by long-time numerical runs
of the dynamics, in order to get a feeling for frequently visited regions of the
phase space,  {\em i.e.} the natural measure,
and to search for close recurrences~\cite{auerbach}.
An initial loop guess $\Loop(0)$ is crafted by taking
a nearly recurring segment of the orbit,
smoothed and made periodic by
a FFT into the wavenumber representation,
dropping the high frequency components, and
an FFT back to the phase space.


\subsection{Future directions}

{\bf (1)  Inertial manifold}:
In devising the \descent\ method we have made a series of restrictive choices,
many of which could be profitably relaxed.
In particular,
the choice of a {\em Euclidean metric \costFct} $\costF[\lSpace]$ has no compelling
merit.
For a flow like the Kuramoto-Sivashinsky,
the low Fourier modes $a_1$, $a_2$, $\dots$ are clearly more important than
the high ones,
$a_k$, $a_{k+1}$, $\dots\;$,  $k$ large.
A more inspired choice would use intrinsic information
about dynamics, replacing $\delta_{ij}F_i F_j$ by
a metric $g_{ij}F_i F_j$
that penalizes straying away in the dangerous, unstable directions
more than deviations in
the tame, strongly contracting ones.
{\em The outstanding challenge and one of the core goals of the
proposal is to find out how to restrict the inversion of the loop gradients
to a low-dimensional inertial manifold subspace.}
Without this, there is no hope of applying the theory to
hydrodynamic turbulence.

\noindent
{\bf (2) Topology}:
 As for high-dimensional flows we are usually clueless as to what the solutions should look like, currently we have no way of telling to which periodic orbit the loop space flow
\refeq{bd5} will take our initial guess, other than to the ``nearest'' periodic orbit of topology
``similar'' to the initial loop.


    We have checked that the \descent\ integration
yields quickly and robustly
the short unstable cycles for standard models of low-dimensional dissipative flows.
A more daunting challenge is a search
for spatio-temporally periodic solutions of PDEs.


\section{Kuramoto-Sivashinsky system}
\label{s:KurSivSyst}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{wrapfigure}{r}{2.4in}
\vspace{-3mm}
\hspace{1ex}
\includegraphics[width=0.35\textwidth]{figs/unfolded.ps}
\hspace{-4ex}
\vspace{-3mm}
\caption{\small\em
The return map $s_{n+1} = f(s_n)$
constructed from 274 periodic solutions~\cite{CCP96}
 of the Kuramoto-Sivashinsky
equations \refeq{ks}, with $s$ the distance
measured along the unstable manifold of the spatio-temporally
periodic solution of the shortest period.
\vspace{-3mm}
}
\label{unfolded}
\end{wrapfigure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
One of the simplest and extensively studied spatially extended dynamical
systems is the Kuramoto-Sivashinsky system\rf{KurSiv} (see Holmes, Lumley
and Berkooz\rf{HLB96} for a delightful discussion of why this system
deserves study),
 \beq u_t=(u^2)_x-u_{xx}-\nu u_{xxxx},
 \ee{ks}
    which arises as an amplitude equation for interfacial
instabilities in a variety of contexts,
such as flame fronts.
Amplitude $u(x,t)$ has compact support, with $x \in [0,2\pi]$ a periodic space
coordinate. The $(u^2)_x$ term makes this a nonlinear system, $t$
 %$t \geq 0$
is the time, and $\nu$ is a ``viscosity'' damping parameter that
irons out any sharp features.
 % The subscripts $x$ and $t$ denote the partial derivatives with respect
 % to $x$ and $t$.
Numerical simulations demonstrate that as
the viscosity decreases (or the size of the system increases),
the ``flame front'' becomes increasingly unstable and turbulent.
The task of the theory is to describe this spatio-temporal
turbulence and yield quantitative predictions for its measurable
consequences.

We have proposed that the Kuramoto-Sivashinsky system \refeq{ks} be used
as a laboratory for exploring the feasibility of the recurrent patterns program.
We now summarize the published
results obtained so far in this direction by Christiansen et al.\rf{CCP96}
and Zoldi and Greenside\rf{ZG96}.
  % \PC{get the parameter from Ronnie}


The solution $u(x,t)=u(x+2\pi,t)$ is periodic on
the  $x \in [0,2\pi]$ interval, so one (but by no means only)
way to solve such equations is
to expand $u(x,t)$ in a discrete spatial Fourier series
  \begin{equation}
 u(x,t)= i \sum_{k=-\infty}^{+ \infty} a_k(t) e^{i k x}
\, .
\label{fseries}
  \end{equation}
     Restrict the consideration to the subspace of odd solutions
$u(x,t)=-u(-x,t)$ for which $a_k$ are real. Substitution of
(\ref{fseries}) into (\ref{ks}) yields the infinite ladder of evolution
equations for the Fourier coefficients $a_k$:
 \begin{equation}
\dot{a}_k=(k^2-\nu k^4)a_k - k \sum_{m=-\infty}^{\infty} a_m a_{k-m}
\,.
\label{expan}
 \end{equation}
    $u(x,t)=0$ is a fixed point of (\ref{ks}), with the $k^2 \nu<1$
long wavelength modes of this fixed point linearly unstable, and the short
wavelength modes stable. For $\nu > 1$, $u(x,t)=0$ is the globally
attractive stable fixed point.
As one varies the burning rate, the flame front
can become very unstable and turbulent:
 Starting with $\nu =1$ the solutions go
through a rich sequence of bifurcations,
% whose detailed knowledge is not
% needed for purpose at hand.
and myriad unstable periodic solutions whose
number grows exponentially with period.

The essential limitation on the numerical studies undertaken so far have
been computational constraints: in truncation of high modes in the
expansion (\ref{expan}), sufficiently many have to be retained to ensure
the dynamics is accurately represented. Christiansen et al.\rf{CCP96} have
examined the dynamics for values of the damping parameter %such as
$\nu=0.029910$, $\nu=0.029924$, close to the onset of chaos, while Zoldi
and Greenside\rf{ZG96} have explored somewhat more turbulent values of
$\nu$.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t!]
%\includegraphics[width=1.8in]{figs/ks11.eps}
%\hspace{0.05in}
%\includegraphics[width=1.8in]{figs/ks12.eps}
%\caption{\small\em
% A Kuramoto-Sivashinsky recurrent pattern projected from
% a $d=32$ Fourier modes truncation onto two dimensions
% picked arbitrarily, here the first two modes
% $\{a_1,a_2\}$.
% Left frame: initial guess $L$. Right frame:
% the periodic orbit $p$ reached by the \descent.
%    }
%\label{f:ks1}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
    With the recently developed variational approach of
\refsect{s:RecurFind} we can already go deeper into the turbulent regime.
A typical initial loop guess is displayed in
\reffig{f:ks1}, along with the periodic orbit
% $p_1, p_2$
found by
the \descent\ method. In a loop discretization (here $N=512$ points represent the loop)
 each point has to be specified in all $d$~dimensions;
 here the coordinates $\{a_1,a_2\}$ are picked arbitrarily.
Such numerical results are only a proof of principle. We do not
know to what periodic orbit the fictitious time flow  will evolve for a
given initial loop, other than that
empirically the flow goes toward the ``nearest'' periodic orbit,
with the largest topological resemblance.

 %
\begin{wrapfigure}{r}{3.0in}
\vspace{-3mm}
\hspace{-1ex}
\includegraphics[width=0.23\textwidth]{figs/ks11.eps}~\includegraphics[width=0.23\textwidth]{figs/ks12.eps}
\hspace{-4ex}
\vspace{-4mm}
\caption{\small\em
 \descent\ at work.
 Left frame: initial loop guess. Right frame:
 the periodic orbit $p$ reached by the \descent\ (the left frame in
\reffig{Fig-recurrent}).
    }
\vspace{-3mm}
\label{f:ks1}
\end{wrapfigure}
%
One pleasant surprise is that even though one is dealing with (infinite
dimensional) PDEs, for strong dissipation values of parameters, the
spatio-temporal chaos is sufficiently weak that the flow can be visualized
as an approximately 1-dimensional Poincar\'e return map $s \rightarrow
f(s)$ from the unstable manifold of the shortest periodic point onto its
neighborhood, see \reffig{unfolded}. This representation makes it
possible to systematically determine all nearby periodic solutions up to a
given maximal period.

So far some 1,000 prime cycles have been determined numerically for
various values of viscosity. %\rf{CCP96,ZG96}.
The differences
between various shortest period recurrent patterns are of the
order of 50\% of a typical variation in the amplitude of $u(x,t)$, so the
chaotic dynamics is already exploring a sizable swath in the space of
possible patterns even so close to the onset of spatio-temporal chaos.
Other solutions, plotted in the configuration space, exhibit the same
overall gross structure. Together, they embody Hopf's vision:
They form the repertoire of the
recurrent spatio-temporal patterns that is being explored by the turbulent
dynamics.

With improvement of numerical codes considerably {\em more
turbulent regimes should become accessible, and will be investigated
within the project proposed here.}

\section{Full Navier-Stokes: plane  Couette turbulence}
\label{s:planeCouetteTurb}

The plane shear flows are among the best studied hydrodynamic
flows, both for the simplicity
of the setting for fundamental studies of hydrodynamic turbulence, and
for the engineering importance of the boundary layer turbulent drag.
Easily visualized, they lend themselves to both colorful visualizations
and colorful phenomenology, see Waleffe\rf{Waleffe2003}.
%
We believe that
turbulent 3-$d$ hydrodynamics is today within reach of the
recurrent patterns program.

An example of what has
already been attained, and what this project
intends to push to a higher level, is the most impressive
application of the recurrent patterns ideas
% to genuine fluid dynamics s
so far, by
Kawahara and Kida~\cite{KawKida01}:
the first
demonstration of existence of an unstable recurrent pattern
in a turbulent hydrodynamic flow.
They
have located an important unstable spatio-temporally periodic solution
in a {\em 15,422-dimensional} numerical discretization of
the three-dimensional
constrained plane  Couette turbulence, with no-slip boundary conditions,
 at $Re=400$.
A 3-$d$ snapshot of this solution at a given
instant is given in \reffig{Fig:KKsnap}.

%
\begin{wrapfigure}{r}{2.4in}
\vspace{-3mm}
\hspace{1ex}
\includegraphics[width=0.35\textwidth]{figs/KawaKida01_05_0001_h.eps}
\hspace{-4ex}
\vspace{-3mm}
\caption{\small\em
A 3-$d$ instantaneous snapshot of an unstable periodic solution embedded in the Couette turbulence~\cite{KawKida01}.
Vectors and colors code velocities and vorticities.
\vspace{-3mm}
}
\label{Fig:KKsnap}
\end{wrapfigure}
%
\refFig{Fig:KKdrag} illustrates why a single unstable periodic solution
might already encode essential information about the turbulent
flow.
In the Couette flow energy is injected locally, through the
drag exerted by the turbulent fluid on the two counter-moving plane walls,
and consumed globally, by the viscous dissipation at small scales.
Plotted horizontally is the energy input $I$,
% I   RLx 0 RLz 0 (@u=@yjy= h + @u=@yjy=+h) dx dz=(2LxLzU=h)
and vertically the dissipation $D$
% D   RLx 0 R+h  h RLz 0 j!j2dx dy dz=(LxLzU2=h)
per unit time for a long-time simulation (green), for the periodic
solution \reffig{Fig:KKsnap} (red), and for a short nearby segment of
a typical trajectory (yellow). Were the dynamics laminar, the solution would
be a stationary point at $(I,D)=(1,1)$. The turbulent solution fluctuates
by almost a factor of 2, with the mean turbulence-induced drag on the
``airplane wing'' substantially larger than what it would have been
for a laminar flow, by a factor of nearly 3.

Periodic orbit theory predicts the time averages and fluctuations of measurable
quantities from their values computed on individual unstable periodic
solutions.
Accurate predictions require sets of the shortest periodic solutions.
What is very encouraging about the Kawahara-Kida
example is that already a {\em single unstable recurrent pattern} yields estimates of the
mean dissipation $D$ and
mean velocity profiles across the Couette
channel with a surprisingly high accuracy, presumably because it happens
to explore the (infinite-dimensional)
phase space region with the highest concentration of
the natural measure.


\section{Theory of recurrent patterns}
\label{s:PerOrsInAction}

%
\begin{wrapfigure}{r}{2.4in}
\vspace{-3mm}
\hspace{1ex}
\includegraphics[width=0.35\textwidth]{figs/KawaKida01_04_0001.ps}
\hspace{-4ex}
\vspace{-3mm}
\caption{\small\em
Frictional drag vs. dissipation for the Couette turbulence.~\cite{KawKida01}
\vspace{-3mm}
}
\label{Fig:KKdrag}
\end{wrapfigure}
%
Suppose that,
armed with a computer and a great deal of skill, one has
determined a set of unstable spatio-temporally periodic solutions:
{\em What} are we to do with them? Here the periodic orbit theory
comes to our rescue, with a
wonderfully counterintuitive surprise:
the theory~\cite{QCcourse} says that
the more unstable the patterns are,  the more accurate will be the
predictions based on a small
number of the shortest recurrent patterns! But as the
short period recurrent patterns have not had sufficient time
to be rendered meaningless by exponential instability,
they are precisely the solutions that can be accurately
(if laboriously) computed.

Now we turn to the
central issue; qualitatively, these solutions demonstrate that the
recurrent patterns can be found, but how is this information to be used
quantitatively?
\PC{credit SRB, implementable}
The {\em periodic orbit theory}~\cite{QCcourse} answers such questions by
assembling individual patterns into accurate predictions for measurable
global averages, such as (let us say) the diffusion of light particles by
turbulent air. The key idea of the periodic orbit theory is to compute
measurable averages by means of a formula which re-expresses the average
as a sum over all the possible patterns grouped hierarchically, by the
likelihood of pattern's occurrence.

Very briefly (for a detailed exposition
consult \refrefs{QCcourse,PG97}),
the task of any theory that aspires to be a theory of
chaotic, turbulent systems is to predict the value
of an ``observable"  $\obser(x)$
from the spatial and time averages evaluated along
dynamical trajectories $x(t)$
% = f^t(x)$
\[
\left<\obser\right> = \lim_{t\to\infty} {1 \over t} \left<\Obser^t \right>
\,,\qquad
\Obser^t (x) = \int_0^t  d\tau \, \obser(x(\tau))
\,.
\]
where x(t) is a point in a high- (and in this context, infinite-)
dimensional state space.  The key idea of the periodic orbit
theory is to extract this average from the leading eigenvalue of
the {\evOper}
$ %\[
{\Lop}^t (x,y) = \delta(y-x(t)) e^{\beta \Obser^t(x)}
$ %\]
via the trace formula\rf{CEflows}
\beq \tr {\Lop}^{t}
     =   \sum_{p}
    \left.
    \raisebox{-4.0ex}[5.5ex][4.5ex]
          {\includegraphics[height=7ex]{figs/traceTube.eps}}
    \right._{p}
     =   \sum_{p} \sum_{r=1}^\infty
               { \period{p} \,
                  \prpgtr{t-r \period{p}}
                  \over \oneMinJ{r} }
                 e^{r \beta \Obser_p}
\label{tr-L1}
\eeq
 which relates the spectrum of the {\evOper} to a sum over prime periodic
solutions $p$ of the dynamical system and their repeats $r$.

What does this formula mean? The intuitive meaning of a trace formula is that
it expresses the average
 $\langle  \exp({\beta \Obser^t}) \rangle$
 as a discretized integral over the dynamical space partitioned
topologically into a repertoire of spatio-temporal patterns, each weighted
by its SRB measure, {\em i.e.}
the likelihood of pattern's occurrence in the long time evolution of
the system.  More precisely: Prime cycles partition the dynamical space
into neighborhoods, each cycle enclosed by a tube whose volume is the
product of its length $\period{p}$ and its thickness
$\left|\det({\bf 1}-{\bf J}_p)\right|^{-1}$. The trace picks up a periodic orbit
contribution only when the time $t$ equals a prime period or its repeat, a
constraint enforced here by $\prpgtr{t- r\period{p}}$. ${\bf J}_p$ is the
Jacobian of cycle $p$, so for long cycles
$|\det({\bf 1}-{\bf J}_p^{r})|\approx$~(product
of expanding eigenvalues), and the contributions of long and very unstable
cycles are exponentially small compared to the short cycles which dominate
trace formulas. The number of contracting directions and the overall
dimension of the dynamical space is immaterial; that is why the theory
also applies to (infinite-dimensional) PDEs. All this information is
purely geometric, intrinsic to the flow, coordinate re-parametrization
invariant, and the same for any average one might wish to compute. The
information related to a specific observable is carried by the weight $
e^{\beta \Obser_p }$, the periodic orbit estimate of the contribution of $
e^{\beta \Obser^t(x)}$ from the $p$-cycle neighborhood.


Periodic solutions are important because they form the {skeleton} of the
invariant set of the long time dynamics, %\cite{cycprl,AACI}, with cycles
ordered {hierarchically}; short cycles give dominant contributions to the
invariant set, longer cycles corrections. Errors due to neglecting long
cycles can be bounded, and for nice hyperbolic systems they fall off
exponentially or even super-exponentially with the cutoff cycle
length\rf{Rugh92}. Short cycles can be accurately determined and global
averages (such as transport coefficients ) can be computed
from short cycles by means of {cycle expansions}.

To emphasize the obvious: the spatio-temporally
periodic solutions are {\em not} to be thought of as eigenmodes, a good
linear basis for expressing solutions of the equations of motion.
Something like a dilute instant approximation makes no sense at all for
strongly nonlinear systems that we are considering here. As the equations
are nonlinear, the periodic solutions are in no sense additive, and their
linear superpositions are not solutions.

In particular, while our approach to turbulence is motivated by the same
considerations as those eloquently explained in Holmes, Lumley, and
Berkooz\rf{HLB96}, the proposed theory is very different.
Holmes {\em et al.}
assume that steady state turbulent dynamics fills out a submanifold of
phase space in an approximately Gaussian-noise way, use the proper
orthogonal decomposition to determine the ``center of mass'' of the
natural measure of the turbulent dynamics, and construct from the spatial
2-point correlation function a finite number of modes description of the
turbulent dynamics restricted to the inertial manifold. The assumption
that the natural measure is concentrated in an ellipsoidal neighborhood of
its center of mass is natural for stochastic processes, but not so for
measures concentrated on fractal sets:  The ``center of mass" of the H\'enon
attractor has little to do with the attractor, for example.


Instead, it is the trace formulas and {\fd s} of the periodic orbit theory
that prescribe how the repertoire of admissible recurrent patterns
is to be systematically explored, and how these solutions are to be put
together in order to predict measurable observables.


\section{Proposed research}
\label{s:PropRes}

PI and collaborators have implemented the recurrent patterns program in
detail~\cite{CCP96} on a relatively simple system, 1-spatial dimension
Kuramoto-Sivashinsky combustion equations that describe the flutter of a
gas flame.
For this specific problem many recurrent patterns
have been determined numerically, and the periodic-orbit theory
predictions tested for various values of flame front parameters.
Kawahara and Kida~\cite{KawKida01} have
demonstrated that the recurrent patterns can be determined in turbulent hydrodynamic flows
by locating an important unstable spatio-temporally periodic solution
in
the three-dimensional
plane  Couette turbulence.

Both advances are a proof of principle, first steps in the
direction of implementing recurrent patterns program. But there is a big conceptual
gap to bridge between what has been achieved, and what needs to be done:
Even the flame flutter has been probed only in its weakest turbulence
regime, and it is an open question to what extent Hopf's vision remains
viable as the system grows large and more turbulent.


\subsection{Collaborative team}

The program outlined above will require full time attention
of one postdoctoral
fellow and one or two Ph.D. students.
The above Kuramoto-Sivashinsky and Navier-Stokes studies are proofs
 of principle, a first
step in the direction of demonstrating the feasibility of the recurrent patterns program.
Implementing the program in fluid dynamics requires
computational power beyond PI's group's expertise.
A project so ambitious will
require that the PI puts together a strong collaborative team
% will seek to break the impasse by bringing together
% a %{\GT} team
from natural sciences (physics, atmospheric sciences, oceanography) and engineering
(aerospace, fluid dynamics) with complementary skills;
experimental, numerical and theoretical.



The  experience and skills of project team's
{\bf computational fluid dynamicists}
are absolutely essential for any further progress.
At Georgia Tech PI intends to ask
{ F.~Sotiropoulos}, Assoc. Professor,
School of Civil \& Environmental Engineering
and
{ P.K.~Yeung},
Assoc. Professor,
School of Aerospace Engineering, to join the
project, and internationally PI is already a consultant
on the current Kawahara-Kida proposal (Japan).
{\bf Experimental fluid dynamics} is pursued by a
experimentalist-theorist team consisting of
{ M.~Schatz},
Assoc. Professor,
and { R.~Grigoriev},
Assist. Professor, School of Physics, and
another experimental biologist-computational fluid dynamicist team
consisting of Professor { Jeannette Yen},
School of Biology, and  F.~Sotiropoulos.


Schatz group possesses a unique skill: by means of their multipoint thermal actuation
technique, they are able to {\em design}
spatio-temporal patterns, and thus create and test experimentally the patterns singled out by the theory. Thermally
actuated control of thin films~\cite{Hexagon} pioneered here relies on
modern electronics being much faster than fluid
dynamics. The Schatz group, with expertise in real-time fluid control,
can both observe~\cite{JY02} and simulate small fluid systems in real time.
Were the theoretical models of comparable sophistication,
this strategy could be reversed, and the real-time simulations could be used to
{\em control} spatio-temporal patterns.

{ J.~Yen}'s team
has the ability to resolve  experimentally 3-$d$ whirls and whorls,
and needs to
quantify the dynamics underlying  interaction
of coherent structures in turbulent flows in real time;
the team will address these challenges relying on experiments
such as the real-time 3D optical measurements
of wakes in fluids, and fluid dynamic simulations
of such wakes~\cite{MezSat02}.



\subsection{Research plan}

To the extent that such plans are ever realistic, here are the proposed research
mileposts:

\begin{enumerate}
\item[Year 1:]
\begin{enumerate}
\item
{\bf Variational searches for recurrent patterns:} Devise
and implement an
inversion of the loop linearized stability restricted
to a finite inertial manifold subspace (regardless of the
dimension of the discretized PDE truncation).
\item
{\bf Kuramoto-Sivashinsky:} Test the variational code on 1-$d$
Kuramoto-Sivashinsky flow for system sizes larger than those
attained so far. The antisymmetric subspace \refeq{expan} is unphysical, so
explore the dynamics on the full space.
\item
{\bf Symbolic dynamics:} Construct symbolic dynamics and
approximate pruning rules for the above
Kuramoto-Sivashinsky flow simulations, construct corresponding
dynamical zeta functions, and compute expectation values of observables.
% \item
% {\bf Intermittency:}
\item
{\bf Plane Couette flow:} commence coding, testing the Navier-Stokes
flow simulation code.
\end{enumerate}

\item[Year 2:]
\begin{enumerate}
\item
{\bf Kuramoto-Sivashinsky:} study equilibrium solutions for the 1-$d$
system of infinite extent. Use this information to classify types
of spatially periodic structures.
\item
{\bf Systems of infinite spatial extent:} implement a
variational search method for recurrent patterns,
compute a set of shortest such patterns in  1-$d$
Kuramoto-Sivashinsky system.
\item
{\bf Plane Couette flow:} reproduce the
Kawahara-Kida unstable solution for plane Couette turbulence,
but now also compute its linearized stability.
\end{enumerate}

\item[Year 3:]
\begin{enumerate}
\item
{\bf Plane Couette flow:} Search for a hierarchy of longer periodic orbits,
and their linearized stability, eigenspectra and and eigenvalues. Compute the
frictional drag for a range of $Re$.
\end{enumerate}


\end{enumerate}

\bigskip


\subsection{Intellectual merit}

We already possess unique tools which we hope will lead to
deterministic predictive capabilities for turbulence more
quantitatively precise and
generally applicable than many of the approaches currently pursued in
fluid dynamics and
engineering literature.
The periodic orbit theory provides the
machinery that converts the intuitive recurrent
 patterns picture into a precise calculational scheme.

The point is, although the initial numerical computation required can be
intensely time, memory  and CPU consuming, once an approximate finite
alphabet of patterns and a vocabulary of realizable short words
(patterns explored sequentially by the dynamics) has been extracted
- and this is done offline -
the result is an alphabet of order
of 100-1000 of patterns.  The dynamics in terms of these patterns might be
fast enough that
it can be implemented in real time - a prerequisite for applying this theory to
engineering problems, such as the turbulent drag control and reduction.


The impact of a major advance here is
by no means restricted to hydrodynamical turbulence.
The key concepts should be applicable to many systems extended in space, and
a successful theory of spatially extended systems would have broad
impact, from problems involving
motions of fluids to subatomic phenomena to assemblies of neurons.
In a parallel
(many degrees-of-freedom) but distinct (Hamiltonian, hence no low-dimensional
attracting sets) PI group's research effort,
NSF pending proposal \#0355349, the semi-classical quantization of
classically turbulent field theories,
classical recurrent pattern solutions will be needed to implement
any Gutzwiller-type trace formula
semi-classical approximation to the quantum theory.

\medskip

%\PC{end with a summary in italics}
%{\bf Summary:}
{\em This proposal will address the grand challenge of nonlinear
  science: Explore experimentally and describe theoretically the dynamics of high-dimensional nonlinear systems. Furthermore,
this proposal will foster applications of the methods thus developed to problems
  in engineering.}

\medskip

In E.~Hopf's own words:
{\em
The ultimate goal, however,
must be a rational theory of statistical hydrodynamics
where
% those important results and other
 [$\cdots$] properties of turbulent flow can be
mathematically deduced from the fundamental equations of hydromechanics.
    } %end {\em

\subsection{Broader Impact}
% from summary.tex          IGERT preproposal

A modern education in the tools and methods of nonlinear science requires
training that bridges traditional discipline boundaries.
Students will acquire both the mathematical tools
and develop physical intuition needed to tackle complex nonlinear problems arising
in many different scientific fields.
The Georgia Tech {\em Center for
Nonlinear Science}~\cite{CNS}  environment will complement the research
component with a broad range of activities:
interdepartmental research seminars, student-run seminars, an active
visiting scientist program, and close interactions with
Georgia Tech groups working on related problems\rf{LGC02,BECprl,BEC-CHAOS},
such as pattern formation and control,
high-dimensional dynamics, coherent structures in turbulent flows.
Collaborative visits to
project partners (C.P.~Dettmann~\cite{Dettm03}
- Bristol, G.~Vattay - Budapest, and
others)
will provide additional training experience and opportunities,
both domestic and abroad.

The outreach initiatives will include undergraduate research participation
and an advanced nonlinear dynamics course.
Currently under development by the {\tt ChaosBook.org}
cross-disciplinary team (particle physicists, mathematicians, nuclear physicists,
condensed matter experimentalists, ...),
this novel hyper-linked web-based advanced
graduate course~\cite{QCcourse}
% available at {\tt www.nbi.dk/ChaosBook}.
is already reaching students across the globe.


\GrantDescr{There are two general merit review criteria approved by the National Science Board (NSB) and listed in the GPG: (1) the intellectual merit of the proposed activity, and (2) the broader impacts resulting from the proposed activity. All proposals must separately address both of the merit review criteria in the Project Summary and should describe the broader impacts as an integral part of the narrative in the Project Description.}

\section{Prior, current and pending support}
\label{s:prior}
\GrantDescr{Achievements Under Prior NSF Support. Describe achievements under
prior NSF support that pertain to the present proposal.
(max {\bf 5 pages})}


% from prior.tex                PFCNS proposal
% PC                                Jan 18 2002

PI's nonlinear dynamics contributions
include the Feigenbaum-Cvitanovi\'c
universal equation for period doubling~\cite{pd}, the theory of
cycle expansions~\cite{QCcourse}, and applications
of the theory to systems that exhibit classical and quantum chaos.
PI is currently Glen Robinson Chair in Nonlinear Sciences and
director of the newly created Georgia Tech {\em Center for
Nonlinear Science} (CNS).

Prior to moving to US, Cvitanovi\'c founded and directed
in the period 1993-1998 {\em Center for Chaos and Turbulence Studies}
(CATS) at the Niels Bohr Institute, Copenhagen, a
cross-disciplinary effort which became one of Europe's leading
centers for nonlinear science, housing and in part funding
%
approximately 15 faculty, 8 post-docs, 45 graduate students, 15
long term visitors, 40 short term visitors, and 5
workshops/conferences in any given year.
%
 As a recent arrival to US, he has no NSF
support. However, in the period 1997-2000, prior to moving to Georgia Tech,
{P.~Cvitanovi\'c} led the initiative to create a Center for
Complex Systems at the Northwestern University, and was the
original PI on the {IGERT \#9987577}: {\em Complex Systems in
Science and Engineering program}, awarded to Northwestern for the
period 2000-2004.


\Preliminary{
\newpage

\noindent{\bf To incorporate}

Hydrodynamic turbulence in the plane shear flows is the natural testing ground,
both at the fundamental level, and
for the engineering importance of the boundary layer turbulent drag.

\bigskip

PI's
group has developed new variational methods for finding recurrent
patterns~\cite{crete03,LanDescent}
and implemented the recurrent patterns program in detail on
1-spatial dimension Kuramoto-Sivashinsky combustion
equations that describe the flutter of a gas flame.  For this specific problem many
recurrent patterns have been determined numerically,
and the periodic-orbit theory predictions tested
for various values of flame front parameters.~\cite{CCP96}

    } % \Preliminary{

\newpage
\setcounter{page}{1}
\label{s:refs}
\input refs

\end{document}
