Biomedical Engineering Reference
In-Depth Information
as long as the original. The deviations from the original simple orbit can grow
larger as the bifurcation parameter is ramped further, eventually leading to a
second period-doubling bifurcation, so that the new orbit has a period that is
four times longer than the original. In fact, an entire period-doubling cascade
can occur within a finite range of the bifurcation parameter, leading finally to a
chaotic attractor. Feigenbaum showed that the sequence of bifurcations has a
structure that is the same for a large class of discrete maps. The details of the
particular map under study become irrelevant as we approach the end of the pe-
riod-doubling cascade.
To see a sequence of period-doubling transitions leading to chaos, one need
look no farther than our simple nonlinear oscillator. Figure 4 shows a set of so-
lutions, with
k
= 1,
k
' = 0.5, H = 0.5, and X = 0.8, for four different values of the
drive amplitude B. The time series on the top row show that bifurcations occur
as the drive amplitude is varied. At some critical point between the first and sec-
ond panels (B = 0.7 and B = 0.76) the solution undergoes a structural change.
It begins with a limit cycle with some period (the exact value is unimportant).
It then changes to a limit cycle that has a period approximately twice as long
as the original. In terms of the original, every other cycle looks different. In a
periodically forced system such as this one, the second limit cycle is often called
a "2:1 state," referring to the fact that there are two periods of the driver
for every one period of the limit cycle. By the time we get to the third panel, the
2:1 state has itself undergone a period-doubling bifurcation, leading to a 4:1
state. In between the third and fourth panels, an infinite sequence of period dou-
blings has occurred, leading, finally, to a strange attractor and its trademark er-
ratic time series.
The bottom row of Figure 4 shows a view of the same motion that clarifies
the nature of the bifurcations a bit. Each of these plots is a projection of the tra-
jectory corresponding to the time series above it. In the present case, the state
space is three dimensional, the three dimensions corresponding to the position
and velocity of the mass and the position of the ceiling. The figures show the
projection of a path through the 3D space onto a 2D plane. Here the differences
between the four solutions are easier to see at a glance.
5
A standard method for analyzing such a situation is through the construction
of a discrete
return map
from its current position in state space to its position
one drive period later. This is the theoretical equivalent of taking a movie of the
motion with a strobe light that flashes in synchrony with the ceiling oscillations.
In that movie, the ceiling will appear to sit still, while the weight will jump from
point to point according to the map. If the weight is on a simple limit cycle as
described above, it will appear fixed in the movie. In this way, we see that a
periodic orbit of a continuous-time system corresponds to a fixed point of a dis-
crete-time system.
6
The motion of the system in the strobed movie is said to oc-
cur on a
Poincaré section
of the state space. For practical and analytical
