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314 chapter 12|θ(t)|200 1 2Figure 12.12 A bifurcation diagram for the damped pendulum with a vibrating pivot (seealso the similar diagram for a double pendulum, Figure 12.14). The ordinate is |dθ/dt|, theabsolute value of the instantaneous angular velocity at the beginning of the period of thedriver, and the abscissa is the magnitude of the driving force f. Note that the heavy line resultsfrom the overlapping of points, not from connecting the points (see enlargement in theinset).fIn the scatterplot in Figure 12.12, we sample ˙θ for the motion of a chaotic pendulumwith a vibrating pivot point (in contrast to our usual vibrating externaltorque). This pendulum is similar to our chaotic one (12.29), but with the drivingforce depending on sin θ:d 2 θ dθ= −αdt2 dt − ( ω0 2 + f cos ωt ) sin θ. (12.43)Essentially, the acceleration of the pivot is equivalent to a sinusoidal variation of gor ω 2 0. Analytic [L&L,M 76, § 25–30] and numeric [DeJ 92, G,T&C 06] studies of thissystem exist. To obtain the bifurcation diagram in Figure 12.12:1. Use the initial conditions θ(0)=1and ˙θ(0)=1.2. Set α =0.1, ω 0 =1, and ω =2, and vary 0 ≤ f ≤ 2.25.3. For each value of f, wait 150 periods of the driver before sampling to permittransients to die off. Sample ˙θ for 150 times at the instant the driving forcepasses through zero.4. Plot the 150 values of | ˙θ| versus f.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 314