Biomedical Engineering Reference
In-Depth Information
Figure 3
. The limit cycle amplitudes for different values of the driving frequency in a nonlin-
ear oscillator. The solid circles correspond to oscillations observed as the driving frequency is
slowly ramped up from 0.7. The open circles correspond to oscillations observed as the driving
frequency is slowly ramped down from 0.9. The dotted lines are guides to the eye. The drive
frequency is measured in units of the natural frequency of the linear oscillator. The time series
shown in Figure 2 correspond to the two limit cycles that coexist at a driving frequency of 0.8
for H = 0.18,
k
= 1,
k
' = 0.5, and B = 0.3.
As mentioned above, nonlinear dynamical systems sometimes exhibit
chaos, motion that never settles into a fixed point or limit cycle. The system
stays confined to a finite region of state space, but never returns precisely to one
of the points it has visited before. In fact, almost all dynamical systems are cha-
otic for some range of parameter values and our simple driven oscillator is no
exception. An example of a chaotic orbit in this system is shown in Figure 4.
Much attention has been devoted to the characterization of strange attrac-
tors. Three aspects of the theory are of particular interest for practical purposes.
The first is the determination of the
dimension of the attractor
. The attractor
itself is a geometric object, a set of points in state space, that has a dimension
which can be non-integral, sometimes called a
fractal dimension
. Most impor-
tantly, the dimension is finite and lower than the dimension of the full state
space. Its origin in a set of deterministic equations for a relatively small number
of variables makes it fundamentally different from the erratic trajectories associ-
ated with random, or stochastic, processes. This raises the possibility that an
experimentally observed time series suggesting erratic, unpredictable behavior
actually arises from a deterministic, though nonlinear, set of equations. A beauti-
