Biomedical Engineering Reference
In-Depth Information
marily in "Hamiltonian" systems, in which conservation of energy prohibits
convergence or divergence of nearby phase space trajectories. Such
dissipa-
tionless
systems are of great interest in quantum mechanics and statistical me-
chanics, but systems of interest in biomedicine always involve strongly
dissipative
processes, which include, for example, all processes involving fric-
tional forces. Marginal stability then occurs only as a very special case where
parameters have been carefully tuned, though there have been suggestions that
marginal stability can reappear spontaneously in certain self-organizing nonlin-
ear systems (2).
Now we are often interested in dissipative systems that are subjected to ex-
ternal driving of some sort, whether it be a steady input of energy or a driving
with more complicated temporal structure. In such systems, the notion of a fixed
point must be generalized to include steady or regularly repeating motions. For
example, if the ceiling from which a weighted, damped spring is hanging were
constantly oscillating up and down, the weight would not sit at a fixed point but
could exhibit regular oscillations with a period that matches the oscillation of the
ceiling. Such trajectories are called
limit cycles
, and, like fixed points, they may
be stable or unstable.
Stable fixed points and limit cycles are called
attractors
, as trajectories in
state space eventually flow toward them and then stay very close to them at long
times. If we begin observing a system when it is far from its attractor and watch
for a long time, we will be able to detect its motion toward the attractor for a
while, but at some point it will be so close to the attractor that we can no longer
resolve the difference. The portion of the trajectory over which we can observe
progress toward the attractor is called a
transient
. The set of points in state
space that lie on transients associated with a particular attractor is called the
ba-
sin of attraction
of the attractor. In a stable linear system, all points in state
space lie in the same basin of attraction. In other words, for any initial configu-
ration of the system variables, the ultimate fate of the system is the same fixed
point or limit cycle.
4.
NONLINEAR EFFECTS IN SIMPLE SYSTEMS
Linear systems are often studied in great detail. They can be solved exactly
and hence make for good textbook problems; and linear equations can be used
as good approximations to nonlinear ones in situations where the trajectories
stay very close to a stable fixed point or limit cycle. They cannot capture, how-
ever, many of the most important qualitative features of real systems.
In a
nonlinear system
, the equations of motion include at least one term
that contains the square or higher power, a product of system variables (or more
complicated functions or them), or some sort of threshold function, so that the
addition of two solutions does
not
yield a valid new solution, no matter how the
