Biomedical Engineering Reference
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Figure 5 . The bifurcation diagram for the oscillator with k = 1, k ' = 0.5, H = 0.5, and X = 0.8,
in the range B = (0.7, 0.82). The dashed vertical lines indicate the values of B used in Figure 4.
fixed point. Though it cannot be seen from these figures, there may still be a
fixed point of f in this system, but it has become unstable at this value of B. For
the chaotic orbit on the right, the sequence x n will contain an infinite number of
different points (assuming we can wait long enough to collect them). Moreover,
there are infinitely many values of m for which f m has a fixed point. These fixed
points cannot be literally on the attractor—if the system were ever to hit one of
these points exactly, it would stay on the associated periodic orbit—but they are
infinitesimally close to it. These periodic orbits are said to be embedded in the
strange attractor, and their presence has been exploited both for control purposes
and for the derivation of mathematical properties of the strange attractors. (See
(24) for details.)
A useful way to exhibit the types of bifurcations that occur in a given sys-
tem is to form a bifurcation diagram from the return map. Sets of values of x n
are collected for many different values of the bifurcation parameter and plotted
on a single figure, as shown in Figure 5. Each vertical slice of the figure shows
all of the x n 's observed for the corresponding value of B. The sequence of period
doubling bifurcations is visible, and is a common structure in systems with only
a few variables. Two other features common to experiments are visible in the
figure. First, at the critical point for the first period doubling bifurcation (near B
= 0.76), the data are slightly smeared out. This is because near the transition the
1:1 limit cycle is just barely stable, which in turn implies that the transient re-
laxation to the limit cycle is very long. The plot was made by integrating the
equations of motion up through about 50 cycles and recording data from the last
40 cycles. In the present case, the smearing could easily be reduced by waiting
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