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\centerline{\tenbf FULLY DEVELOPED CHAOS}
%\centerline{\tenbf A DYNAMICAL SYSTEM WITH SPATIAL COMPLEXITY}
%\centerline{\tenbf INCREASING IN TIME}
\vglue 1.5cm
\centerline{\tenrm P.COLLET}
\baselineskip=13pt
\centerline{\tenit Centre de Physique
Th\'eorique\footnote"${}^{1}$"{\ninerm Laboratoire CNRS UPR 14}}
\centerline{\tenit Ecole Polytechnique}
\centerline{\tenit F-91128 Palaiseau Cedex (France)}
\vglue 1.0cm
\centerline{\tenrm ABSTRACT}
{\rightskip=3pc\leftskip=3pc\tenrm\baselineskip=12pt\parindent=1pc
We describe a non linear partial differential equation on a compact
domain with solutions regular but with rough behaviour down to a small
scale.
\vglue 0.5cm}
\line{\elevenbf 1. Introduction\hfil}
\vglue 0.5cm
\baselineskip=14pt\elevenrm
The aim of this paper is to present a simple model exhibiting the
development of a multifractal structure through a dynamical evolution.
There are many physical systems with an apparent complexity
which increases in time like for example the dendritic growth of a solid,
the DLA dynamics, the Safmann-Taylor fingers, interfaces etc.
One of the most spectacular example being the snowflakes.
The phenomenon of spatial intermittency in turbulent flows can also be
interpreted as a manifestation of singularities at some scale\ref{1,2}, and the
discussion below will be partly inspired by this example.
All these systems evolve
in time according to some known dynamical equations and develop
structures of increasing complexity (sometimes called turbulent
structures).
During a long transient period,
the complexity of theses systems does not seem to be
stationary, and the apparent roughness of the solutions seem to invade
smaller and smaller scales down to a physical cut-off (diffusion length,
capillarity length etc.).
One of
the difficulties in understanding theoretically these complex structures
is that the dynamical evolution of the different spatial scales
are intricately coupled. One would like to understand why these
time evolutions which are given by partial differential equations
(supplemented in some problems with moving
boundaries) produce these apparently irregular structures. Such questions
are rarely considered in the Mathematics literature where one
finds a lot of information about existence, uniqueness and regularity of
solutions or the opposite case of the development of a singularity in
finite time.
In order to gain a better understanding of the
phenomenon we study a simple model whose time evolution is given by a
partial differential equation. The solution is unique, bounded and
regular at any finite time. However it develops a spatial roughness
which increases in time and invades smaller and smaller scales.
Our aim is to discuss this simple model
which can be understood without too much technical difficulties.
As we will see, the behaviour seems to be qualitatively similar
to the more realistic time evolutions mentioned above, although the
important effect of the coupling between the different
scales remains to be understood.
We emphasize that
the models discussed below have deterministic time evolutions, and their
complexity is coming from an intrinsic stochasticity. This is a
different situation from the models
with stochastic forcing which have been previously studied (see for
example\ref{3}).
The so called turbulent transport of a passive quantity is a phenomenon
related to the questions discussed in the present paper\ref{4}. However
the complexity is again of an exterior (though deterministic) origin:
the chaotic properties of the transporting flow.
\vglue 0.5cm
\line{\elevenbf 2. The model\hfil}
\vglue 0.5cm
We now describe our model by giving first the phase space and then the
time evolution. We will only describe a very simple version of the
idea, many other models can be developed along the same lines.
We consider as physical base space a circle where the points are indexed
by an angle $\theta$. This angle will be the space variable of the model.
At each spatial location we have a local order parameter which is a
three dimensional vector. Therefore, the phase space of our system will
be the space of periodic vector valued functions on the circle
$$
\theta\to v(\theta)\;.
$$
This is
to be compared with the case of a fluid motion where the phase space is
the set of vector fields (velocity fields) in the container with the
appropriate boundary conditions or with the case of a growing closed curve
given parametrically. We now introduce the time evolution, first in
degenerate form. This time evolution will determine the
configuration $v(\theta,t)$ of the system at time $t$ given the
configuration $v(\theta,0)$ of the system at time $0$. It is given by
(a partial differential equation)
$$
\partial_{t}v(\theta,t)=X(v(\theta,t))\;, \eqno(1)
$$
where $X$ is the vector field of the Lorenz model with parameter values
in the chaotic regime, for example
$$
\eqalignno{X_{1}(x_{1},x_{2},x_{3})=&10(x_{2}-x_{1})\;,\cr
X_{2}(x_{1},x_{2},x_{3})=&28x_{1}-x_{2}-x_{1}x_{3}\;,&(2)\cr
X_{3}(x_{1},x_{2},x_{3})=&-8x_{3}/3+x_{1}x_{2}\;.\cr}
$$
To be rigorous, one should take instead a geometric Lorenz flow for
which chaos has been proven to occur\ref{5,6}. In order to simplify the
exposition we will however continue to use the Lorenz system.
The above evolution equation may look surprising at first sight in
particular because it does not contain terms with partial derivatives in
$\theta$ (although we want to view this equation as a partial
differential equation in time and space).
This is however an interesting simplification which allows for
an easy discussion of the main phenomenon. It is at
least formally reminiscent of the Eulerian limit of the Navier-Stokes
equation with a vanishing viscosity.
Let us consider an initial condition $v(\theta,0)$
for our system, that is to say a
periodic vector valued function. Let us assume that this function is
very regular (infinitely differentiable), and not constant.
After a period of time $t$ has elapsed, the configuration is a function
$v(\theta,t)$ which is still periodic in the angle $\theta$. It is easy
to verify that this function is also infinitely differentiable. However
down to a certain scale it has in a certain sense very large derivatives.
This can be seen more precisely as follows. Fix a small number $\epsilon$
and consider the variation of the function on an interval of length
$\epsilon$. This is given by
$$
v(\theta+\epsilon,t)-v(\theta,t)\sim \epsilon \hbox{\rm D}\phi_{t}(v(\theta,0))
\partial_{\theta}v(\theta,0)
$$
where $\phi_{t}$ is the flow of the Lorenz vector field (2):
$$
\partial_{t}\phi_{t}(\cdot)=X(\phi_{t}(\cdot))\;,
$$
and $\hbox{\rm D}\phi_{t}$ is the differential of this map. We see from
this formula that due to the sensitivity to initial condition (or
positive Lyapunov exponent) of the Lorenz system, for a fixed number
$\epsilon$, if we wait long enough, the increment of the configuration
on an interval of length $\epsilon$ will become of order 1. Of course,
if on the other hand we fix the time and consider a scale which is much
smaller than the expansion factor, we see regular increments since the
configuration is always infinitely differentiable. In other words, small
variations of the angle $\theta$ induce small variations of the vector
field $v$, but these small variations are exponentially amplified during
the time evolution.
Note also that the space derivative of $v$ exhibits spatial
intermittency. Indeed, we expect that for most values of $v(\theta,0)$,
the vector
$$
\hbox{\rm D}\phi_{t}(v(\theta,0))\partial_{\theta}v(\theta,0)
$$
will be of the order of $e^{\lambda t}$ where $\lambda$ is the Lyapunov
exponent. However this result is in general not true everywhere (except
for very uniform systems), and we expect a larger result ($e^{\alpha t}$
with $\alpha > \lambda$) on sets of Hausdorff dimension smaller than 1
in the one dimensional manifolds of $v(\theta,0)$ (or $\theta$) variable.
As mentioned before, this model does not exhibit any coupling between
the different angles $\theta$ (our space variable).
One can add to the evolution equation a coupling
in the form of a small diffusive term. The evolution equation is now
$$
\partial_{t}v(\theta,t)=\eta\partial_{\theta}^{2}v(\theta,t)+X(v(\theta,t))\;.
\eqno(3)
$$
This equation is not as simple to discuss rigorously as the one before,
but it is not very difficult to prove that regular initial conditions
will evolve into uniformly bounded and uniformly regular solutions.
This follows by standard energy-like estimates.
However, if the diffusion constant $\eta$ is small, one gets a rather rough
function down to a scale ${\cal O}(\eta^{1/2})$ if time is large enough.
Another variant of the model is obtained by using a vector field
different from the Lorenz system. One can for example use a dynamical
system with intermittent properties and this will lead to more space-time
intermittency.
\vglue 0.5cm
\line{\elevenbf 3. Quantitative results\hfil}
\vglue 0.5cm
As we have seen above, the model described by equation (1) develops roughness at
small scale. We now briefly describe the multifractal properties of the
solutions by looking at the local variation of one of the component.
This local variation can be defined in terms of a local H\"older
exponent. One possible definition for a real function $g(\theta)$ is
(see Hardy\ref{7} for a different one)
$$
\alpha(\theta)=\lim_{\epsilon\to0}{\log|g(\theta+\epsilon)-g(\theta)|
\over\log \epsilon}\;.
$$
It is easy to generalize this formula for vector valued functions.
The associated $f(\alpha)$ function would measure as usual the Hausdorff
dimension of the level sets of the function $\alpha$.
As it is now well known, a powerful method for dealing with the
function $f(\alpha)$ is the thermodynamic
formalism. Here we can adapt the analysis to the fact that our order
parameter is of dimension 3 by defining the partition function by
$$
Z_{n}(\beta)=\sum_{k=0}^{2^{n}-1}\|\phi_{t}(v((k+1)2^{-n}))-
\phi_{t}(v(k2^{-n}))\|^{\beta}\;.
$$
The pressure function $P(\beta)$ is defined by
$$
P(\beta)=\lim_{n\to\infty}n^{-1}\log Z_{n}(\beta)\;,
$$
if the limit exists. Under some technical conditions, the function
$f(\alpha)$ is then the Legendre transform of $P$.
The above expression for $Z_{n}$ can be simplified by using Taylor's
formula. We get
$$
Z_{n}(\beta)\simeq 2^{-n(\beta -1)}\int_{0}^{2\pi}\|\hbox{\rm
D}\phi_{t}(v(\theta,0))\partial_{\theta}v(\theta,0)\|^{\beta}\,d\theta\;.
$$
Strictly speaking, this formula gives a good approximation for
$Z_{n}(\beta)$ only if the remainder in the Taylor formula can be
neglected. This is certainly the case if $n$ is large enough. The
integral above looks very much like the partition function of the Lyapunov
exponent. The measure is not however the invariant measure of the Lorenz
system but the Lebegue measure on the curve $v(\cdot,0)$.
However, if the curve meets
the unstable foliation on a set of positive Lebegue measure, one gets
asymptotically for large time the partition function of the Lyapunov
exponent of the SRB measure (see Bowen\ref{8}
for a definition). We therefore obtain
$$
P(\beta)\simeq 1-\beta+{t\over n}P_{L}(\beta)\;,
$$
where $P_{L}(\beta)$ is the pressure of the Lyapunov exponent for the
SRB measure. From this formula, we see that if $n\gg t$ we get a trivial
result corresponding to the fact that the solution is regular at very
small scales. However, if we take a diagonal procedure, letting $n$ diverge
with $t/n$ fixed, we get a non trivial result. The formula gives of
course a good approximation only if $t/n$ is small, because otherwise
the remainder term in the Taylor formula cannot be neglected anymore.
In particular, it is easy to see that if $t/n$ is larger than the
minimal expansion rate of the system, we get $P(\beta)=1$.
One would like of course to get a result in the infinite time regime.
Such a result could be obtained using an invariant measure for the
dynamical system. It turns out that in the case of zero diffusive coupling
one can construct easily an invariant measure.
It is the continuous product\ref{9} of the
Lorenz SRB measures on the circle. The support of this measure seems
however to be very large, and it is not known if the pressure is defined
for almost every curve with respect to this measure.
The introduction of a diffusive coupling should improve the situation by
producing a more reasonable invariant measure.
One can then ask if there is a diagonal limit obtained by letting $n$
diverge with $n/\log\eta$ fixed which is almost surely independent of
the initial condition chosen at random according to this reasonable measure.
\vglue 0.5cm
\line{\elevenbf 4. References \hfil}
\vglue 0.5cm
\item{1.} R.Benzi, G.Paladin, G.Parisi, A.Vulpiani, J. Phys. A
{\elevenbf 17}, 3521 (1984).
\item{2.} U.Frisch, G.Parisi, in {\elevenit Turbulence and
Predictability in Geophysical Fluid Dynamics and Climate Dynamics.}
M.Ghil, R.Benzi and G.Parisi ed. (North-Holland, New York 1985).
\item{3.} L.Bunimovitch, Ya.G.Sinai, in {\elevenit Nonlinear Dynamics and
Tur\-bu\-lence.} G.I.\-Ba\-ren\-blatt, G.Iooss, D.D.Joseph ed. (Pitman, Boston
1983).
\item{4.} E.Ott, T.Antonsen, Phys. Rev. {\elevenbf A44}, 851
(1991).
\item{5.} J.Guckenheimer, P.Holmes, {\elevenit Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields} (Springer, Berlin,
Heidelberg, New York 1983).
\item{6.} M.Kardar, G.Parisi, Y.C.Zhang, Phys. Rev. Lett. {\elevenbf
56}, 889 (1986).
\item{7.} G.H.Hardy, Trans. Amer. Math. Soc. {\elevenbf 17}, 301 (1916).
\item{8.} R.Bowen, {\elevenit Equilibrium States and the Ergodic Theory
of Anosov Diffeomorphisms}, Lecture Notes in Mathematics 470
(Springer, Berlin, Heidelberg, New York 1975).
\item{9.} A.Guichardet, Commun. Math. Phys. {\elevenbf 5}, 262 (1967).
\bye