Biomedical Engineering Reference
In-Depth Information
and destroyed by propagating wavefronts. As one might expect, there is a whole
zoo of observed patterns and bifurcations in such systems, obtained both
from physical experiments and numerical simulations on systems as diverse
as vertically vibrated layers of sand, layers of fluid heated from below, chemi-
cal reaction-diffusion systems, and optical systems involving broad laser
beams in feedback loops containing nonlinear elements. Studies of such sys-
tems appear to be relevant for explaining pattern formation on butterfly wings,
cardiac alternans and fibrillation, and the behavior of neuronal tissue, to name
just a few examples.
Not all spatially extended systems show dynamics qualitatively different
from simple systems of a few variables. Typically, there is a length scale associ-
ated with the spatial patterns one sees in a snapshot of the system. This could be,
for example, the average width of stripes observed in a stationary or moving
pattern or the size of a square in a checkerboard pattern. If the system is not too
large compared to this characteristic length, the dynamics generally takes the
forms discussed in section 3.
To analyze spatiotemporal dynamics, one often tries to define new variables
that make the problem as simple as possible. These variables take the form of
spatially varying functions of the natural field variables, and these function are
called
modes
of the system. Choosing a useful set of modes can be difficult,
though in some cases symmetry considerations make the task easier. For exam-
ple, when the equations of motion are unchanged by uniform spatial shifts, it is
often useful to use a Fourier decomposition, in which the modes is are simple
sine waves of different wavelength. In other cases it may be natural to define
modes associated with spatial structures whose amplitudes grow or shrink par-
ticularly rapidly or capture salient features of the observed patterns.
The partial differential equations of motion are then transformed into ordi-
nary differential equations governing the amplitudes of the different modes. In
the case of systems that are not too large compared to their characteristic length
scale, one usually finds that all but a few of the mode amplitudes decay rapidly
to zero. The long time dynamics of the system is then well represented by cou-
pled ordinary differential equations for a few variables and the methods of sec-
tion 3 can be applied even though the corresponding spatiotemporal behavior
may look rather complicated.
If the system is
large
compared to the natural length scale of the spatial
pattern, the situation becomes substantially more complex. Figure 6 shows a
snapshot a the convection pattern in a fluid heated from below. PDEs used to
model this system reproduce the observed behavior very well, and the phenome-
non is now known as "spiral defect chaos" (7,19). The spiral structures in the
pattern move around in erratic ways, and theoretical understanding of the motion
is far from complete. It can be extremely difficult, for example, to answer one of
the most fundamental questions about observed erratic behavior: Does it corre-
spond to a strange attractor or just to an extremely long transient?
