Consider the chaotic motion of a DDP for which the Liapunov exponent is λ = 1, with time measured in units of the drive period as usual. (This is very roughly the value found in Problem 12.13.) (a) Suppose that you need to predict ∅ (t) with an accuracy of 1/100 rad and that you know the initial value ∅ (0) within 10-6rad. What is the maximum time tmax for which you can predict ∅ (t) within the required accuracy? This tmax is sometimes called the time horizon for prediction within a specified accuracy. (b) Suppose that, with a vast expenditure of money and labor, you manage to improve the accuracy of your initial value to 10-9radians (a thousand-fold improvement). What is the time horizon now (for the same required accuracy of prediction)? By what factor has tmax improved? Your results illustrate the difficulty of making accurate long-term predictions for chaotic motion.
You can see in Figure 12.13 that for y = 1.105, the separation of two identical pendulums with
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