Biomedical Engineering Reference
In-Depth Information
Dynamical systems are, by definition, deterministic. They are therefore ca-
pable of exhibiting exquisitely detailed mathematical structures, and one might
well ask whether these structures survive in the presence of stochastic influ-
ences, or
noise
. Conversely, dynamical systems with rather mundane behavior
could respond in unexpected ways in the presence of noise. The theory of noisy
dynamical systems, which in many cases is studied under the heading of non-
equilibrium statistical mechanics, is a rich topic in its own right and is likely to
be highly relevant for understanding some biomedical processes. It is also true,
however, that the effects of noise can often be safely neglected, either because
the details washed out by the noise are on such a fine scale as to be uninterest-
ing, or because the feedback elements in the system allow it to operate reliably
even when noise is a strong influence.
In spatially extended dynamical systems, boundary conditions can play a
crucial role in determining the nature of the solutions to equations describing a
bulk material. The same PDE can exhibit very different solutions when the
boundary conditions are changed, and the realistic modeling of a system may
depend just as much on getting the boundary conditions right as it does on mod-
eling the bulk process. This often means having to understand the physics of a
material or interface that was originally thought to be external to the system.
Many analytical and numerical studies of PDEs are performed on domains that
are artificially modeled as having no boundaries, like a torus. This is often quite
useful, but care must be taken in applying intuition from these studies to the
interpretation of experiments.
Phase locking
is a phenomenon that occurs when two autonomous systems
that oscillate at different natural frequencies are weakly coupled. While for ex-
tremely weak coupling there exist
quasiperiodic
trajectories of the coupled sys-
tem that never exactly repeat but do not have the positive Lyapunov exponents
associated with chaos, slightly stronger coupling tends to cause the two original
systems to lock into a periodic trajectory in which the ratio of the periods of
oscillation of the two original systems is a rational number. The most famous
case of this is the phase locking of the moon's rotation about its axis to its orbit
around the earth, which is why we on earth always see the same side of the
moon. When elements are added to a system to induce phase locking, or when a
large number of systems become phase locked in a 1:1 pattern, the phenomenon
is sometimes called
synchronization
. In studying natural systems where syn-
chronization is observed, it may be helpful to keep in mind the fact that it could
be a straightforward consequence of nonlinear dynamics principles (29).
Finally, in an age in which the control and manipulation of biological sys-
tems is attracting so much interest and speculation, it is worth noting that there
is a vast and growing literature on the
control
of dynamical systems. In this con-
text, control means applying signals, hopefully of low power, in order to get a
system to follow a desired trajectory in state space. (Two useful textbooks for
basic elements of control theory are (21,22).) This may mean steering the system
