Biomedical Engineering Reference
In-Depth Information
Figure 1
. A simple nonlinear dynamical system. Left: A mass is attached to the ceiling by a
spring. The force exerted on the mass by gravity and the spring together is
k
(
h
-
x
) +
k
'(
h
-
x
)
2
,
where
h
is the displacement of the ceiling from its nominal height and
x
is the displacement of
the mass from its resting position. The solid and dotted images represent the spring and mass at
different times during a cycle in which the ceiling is oscillating. Right: Two time series for the
linear case
k
' = 0.
Figure 1 also shows the behavior of the mass when
k
' = 0, which makes the
system linear. Two time series are shown for a particular choice of the drive
frequency X, and one sees that the long term behavior in the two cases is identi-
cal. The difference between the two curves in the early stages corresponds to
transients that depend on the details of the initial configuration. The behavior
that is reached in the long term is called a limit cycle attractor.
When
k
' = 0, one can see immediately from the equations that the strength
of the drive, B, is not an important parameter in determining the qualitative
structure of the motion. The solution for a given B can simply be rescaled by
multiplication so as to correspond to a different value of B.
When
k
' is nonzero, so that the system is nonlinear, one often finds behavior
similar to that shown in Figure 1, i.e., convergence of all transients (in the do-
main of initial conditions of interest) to the same solution. In the nonlinear case,
however, it is possible to see quite different behavior. Figure 2 shows one sim-
ple nonlinear effect: one can have two different long term solutions for a single
value of the system parameters. The differences produced by different initial
conditions in this case are not limited to transient effects. This phenomenon of
bistability
is a generic feature of nonlinear dynamics, and its presence in all
sorts of biomedical systems indicates that nonlinearities play a fundamental role
in their function.
The presence of bistability in a system raises the question of which initial
conditions will lead to which orbit, or which points in state space lie in which
basin of attraction. Even for systems as simple as the damped, driven oscillator,
