Biomedical Engineering Reference
In-Depth Information
Figure 6
. Spiral defect chaos: an example of spatiotemporal chaos in a system that is large
compared to the characteristic length scale of the pattern (the width of the stripes). Image cour-
tesy of G. Ahlers.
The theory of spatiotemporal pattern formation and chaos in large systems
spans several loosely connected approaches. One is to study the dynamics of
isolated typical structures. For example, one can study the speed with which a
single pulse moves and spreads through an otherwise homogeneous medium.
Another is to look carefully at critical points where universal bifurcation struc-
tures can be identified. An example of this is the onset of stripe structures in a
homogeneous medium, such as the regions of sinking and upwelling in a fluid
heated from below, or the development of chemical patterns in reaction-
diffusion systems. Still another is to analyze the
short-time Lyapunov expo-
nents
and associated modes of instability. These exponents do not quite have the
same meaning as the true Lyapunov exponents, but are similar in spirit. The true
exponents are defined as global properties of the full limit cycle or strange at-
tractor. The short time exponents describe the local stability properties of a tra-
jectory over a finite time interval and the modes associated with positive short-
time exponents can reveal the locations in a pattern where instabilities will make
prediction difficult over near term.
Still another theoretical approach to complex spatiotemporal behavior is to
identify local structures that control the evolution of the pattern and try to de-
scribe their collective behavior in statistical terms. Often, the objects of interest
are "defects" in an otherwise regular pattern. In spiral defect chaos, for example,
it is known that there is another attractor consisting of uniform stripes, so it is
tempting to think of the core of a spiral can be thought of as a defect in a stripe
pattern. For topological reasons, the defect (a left-handed spiral, say) cannot be
