Phys. 4267/6268Assignment 11Problem 1Do 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 whenwe 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 exponentλ1determines the average time rateof divergence of two trajectories, i.e.,λ1= limt→∞1tln|η(t)||η(0)|,whereη(t) =x2(t)-x1(t) is an infinitesimal vector.Problem 2Do problem 9.3.8 from Strogatz.Problem 3To illustrate the ”time horizon” after which the prediction for chaotic dynamics becomes impossible, numericallyintegrate the Rossler system˙x=-y-z,˙y=x+ay,˙z=b+z(x-c)fora=b= 0.2 andc= 5.(a) Starting at two initial conditions separated by a vectorη(0) with an arbitrary direction and length 10-6, computeand plot the separation|η(t)|between trajectories as a function of time. How long does it take forη(t
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