Biomedical Engineering Reference
In-Depth Information
Figure 2
. Two time series for the same system as in Figure 1, but with parameters set such that
two different long time solutions exist. Which solution is realized depends upon the choice of
initial conditions.
the boundaries in state space of the basins of attraction can be quite complex,
making it extremely difficult to predict which orbit the system will eventually
reach from a specified initial configuration. (See chapter 5 of (24) for details.) In
studying more complicated systems, one often finds multiple attractors with
basins of attraction that can vary widely in size.
A feature generically associated with bistability is
hysteresis
, the depend-
ence of the observed solution on the direction in which a parameter is varied.
For example, Figure 3 shows curves indicating the amplitude of oscillation of
the mass in our simple model as the drive frequency X is slowly ramped up and
then down. For small X and large X there is only one attractor. In the intermedi-
ate range, however, there are two (plus an unstable periodic orbit that is not
seen). During the upsweep, the system stays in the basin of one of the attractors
until that attractor is destroyed, at which point it is attracted to the stable orbit of
significantly different amplitude. During the downsweep, the same process hap-
pens in reverse, except that the jump occurs at a lower value of X.
The jump to a different solution in a hysteretic system is an example of a
bifurcation
. More generally, the theory of bifurcations describes the transitions
that occur between structurally different solutions as a system parameter is var-
ied. Such transitions may correspond to the creation or destruction of fixed
points or simply to changes in the stability properties of existing fixed points. In
the oscillator example, one may observe a bifurcation upon variation of any of
the parameters
k
,
k'
, H, B, or X. The precise values of the parameters at which a
bifurcation occurs are called a
critical point
in parameter space. The mathe-
matical theory of how solutions can be created or destroyed as parameters are
varied is well developed and full of beautiful structures (12,24,25,28).
