# Spatiotemporal Chaos

Although a great deal is now known about low dimensional chaos - the erratic motion of
dynamical systems described by a few variables - much less is understood about systems
where the number of chaotic degrees of freedom becomes very large. One example of such
a system is a spatial array of coupled chaotic elements. Typically such sytems show disorder
in both space and time and are said to exhibit **spatiotemporal chaos**. We might
hope that a *statistical* description of a very *large* (or infinite) number of degrees
of freedom could be simpler than a *detailed* description of an intermediate number
(say 10-100) of degrees of freedom.
At the present stage of our knowledge, it is important to be guided in our theoretical attack
by the experimental phenomena. We have therefore been studying, using numerical
simulations of equations that model the fluid dynamics, systems that are also being
investigated experimentally.

The research leading to the results presented here was supported by the NSF.

### Spiral Chaos

__Spiral Chaos__. (Click for 200 kbyte mpeg movie
or __here__ for a longer 1.2 Mbyte movie)

These are results from __numerical simulations__ of a model
for __Rayleigh-Benard Convection__
Spiral choas in convection was discovered experimentally by Morris, Bodenshatz, Cannell and
Ahlers. A description of some experimetal results can be found
__here__.
Although dynamic spiral states are familiar in chemical and biological systems where the
underlying instability is to waves, the state was completely unexpected in Rayleigh-Benard
convection where the instability is to a stationary stripe state (the convection rolls). The state
is even more intriguing since straight parallel rolls are thought to be stable at the same parameter
values at which the dynamic spiral state is seen.

We have attempted to explain the existence of the state in terms of what we call *invasive
defects*. A copy of the
__ paper__
is on the Los Alamos preprint library..

### Domain Chaos

If a convection apparatus is rotated about a vertical axis, the Coriolis forces change the fluid motion.
For large enough rotation rates the stationary stripe pattern becomes unstable to a dynamic pattern
of carniverous domains i.e. patches of rolls at one orientation eat up neighboring patches at some
other orientation, in turn to be eaten by a third set of patches e.t.c.
These are results from __numerical simulations__ of a model
for *rotating* __Rayleigh-Benard Convection__

__Domain Chaos__. (Click for 260 kbyte mpeg movie)

We can look at the same dynamics in a number of different representions.

__Stripe Orientation__. (Click for 340 kbyte mpeg movie)

To make the domains clearer we can simply show the orientation of the stripes at each point,
here plotted using a circular rainbow color scale from 0 to 180 degrees.

__Domain Walls__. (Click for 160 kbyte mpeg movie)

Alternatively we cal look at the domain wall motion - extracted from the data as regions where
the amplitude of the stripe pattern is supressed.

You can also see longer __domain__ (1400 kbyte) or
domain wall (650 kbyte) movies (more of the same!).

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The research leading to the results presented here was supported by the NSF.

Last modified Monday, March 24, 1997

Michael Cross