[Computer] Since the rolling motion of Problem 12.17 is periodic (and hence not chaotic) we would…

[Computer] Since the rolling motion of Problem 12.17 is periodic (and hence not chaotic) we would expect the difference ∆ϕ (t) between neighboring solutions (solutions of the same equation, but with slightly different initial conditions) to decrease exponentially. To illustrate this, do part (a) of Problem 12.17 and then find the solution of the same problem except that ϕ (0) = 1. Call this second solution ϕ2 and let ∆ϕ (t) = ϕ2 (t) – ϕ1 (t) . Make a plot of log I ∆ϕ (t)I against t and comment.

Problem 12.17

[Computer] In Figure 12.15, you can see that for γ = 1.503 the DDP “tries” to execute a steady rolling motion changing by 2π once each cycle, but that there is superposed an erratic wobbling and that the direction of the rolling reverses itself from time to time. For other values of γ, the pendulum actually does approach a steady, periodic rolling. (a) Solve the equation of motion (12.11) for a drive

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Figure 12.15

 

 

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