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# hw11 - Phys 4267/6268 Assignment 11 Problem 1 Do problem...

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Phys. 4267/6268 Assignment 11 Problem 1 Do problem 9.3.1 from Strogatz. I haven’t formally introduceded the Lyapunov exponents in class yet, but you have already seen one of them when we discussed the exponential sensitivity of chaotic systems to initial conditions. Lyapunov exponents are the general- ization of the Floquet exponents to aperiodic trajectories. The largest exponent λ 1 determines the average time rate of divergence of two trajectories, i.e., λ 1 = lim t →∞ 1 t ln | η ( t ) | | η (0) | , where η ( t ) = x 2 ( t ) - x 1 ( t ) is an infinitesimal vector. Problem 2 Do problem 9.3.8 from Strogatz. Problem 3 To illustrate the ”time horizon” after which the prediction for chaotic dynamics becomes impossible, numerically integrate the Rossler system ˙ x = - y - z, ˙ y = x + ay, ˙ z = b + z ( x - c ) for a = b = 0 . 2 and c = 5. (a) Starting at two initial conditions separated by a vector η (0) with an arbitrary direction and length 10 - 6 , compute and plot the separation | η ( t ) | between trajectories as a function of time. How long does it take for η ( t
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