Biomedical Engineering Reference
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Figure 4 . The period doubling route to chaos in the oscillator with k = 1, k ' = 0.5, H = 0.5, and
X = 0.8 for B = 0.700, 0.760, 0.794, and 0.810. Top: time series of the position ( x vs. t ). Bot-
tom: phase space plots ( x vs. x ).
reasons, one often works directly with a discrete map that takes one point on the
Poincaré section into the next, rather than the underlying differential equations.
Since the system is deterministic, the map that takes one point to the next is
unique. In the oscillator example above, the Poincaré section may be taken to be
the half plane corresponding to the points where the phase of the drive has some
chosen value. The dashed line drawn on the lower set of plots in Figure 4 sche-
matically represents the projection of this plane onto the x - x plane. As time
progresses, the system keeps looping around the state space in a clockwise di-
rection, passing through the Poincaré section once every time around. Each time
the section is crossed, the position and velocity of the oscillator are observed. In
the present case (and many others) it is sufficient to keep track of only one vari-
able, say the position at each piercing of the Poincaré section. In this way we
obtain a discrete sequence of x values, x n . A return map f defined by x n +1 = f ( x n ;B)
can then be constructed, where we write B explicitly to indicate that the map
depends on the bifurcation parameter.
For the simple limit cycle on the left, the system returns to the same point
on every cycle. For this value of B, the fixed point x * satisfying x * = f ( x *) is
stable. For the second case shown in the figure, x n will alternate between two
values. In this case it is the map f 2 (two successive applications of f ) that has a
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