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dynamical-systems-hw13

dynamical-systems-hw13 - trajectories estimate the largest...

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Instructor: Wayne Hacker Math 550.391-f08 Homework Set 13: ODEs on the real line Grading Scheme for homework problems: Each problem will be graded according to the following scheme: Minor algebra/calculus mistake with a correct approach: 2 out of 3 points given Major algebra/calculus mistake with a correct approach: 1 out of 3 points given Wrong approach: 0 points given The differences between major and minor mistakes will be determined by the instructor, not the student! This algorithm will be strictly enforced. No exceptions! Difficulty Rating Scale: Next to each problem will be the following labels. E = Elementary Level Problem I = Intermediate Level Problem D = Difficult Level Problem

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Math 550.391, Homework Set 13-f08, Hackernotes c Wayne Hacker 2008. All rights reserved. 2 The Problems for Chapter 9: Problem 1: (rating = E) Strogatz 9.2.1 Problem 2: (rating = E) Strogatz 9.2.2 Problem 3: (rating = I) Strogatz 9.3.1 Problem 4: (rating = I) (Strogatz 9.3.9) Using numerical integration of two nearby
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Unformatted text preview: trajectories, estimate the largest Liapunov exponent for the Lorenz system, assuming that the parameters have the values r = 28, σ = 10, b = 8 / 3. You will need to use the 4th-order Runge-Kutta integrator ode.m on the course website, since the matlab integrators like ode45 do not use a uniform time-step. Type [ y1 , t ] = ode(D, y0 , t0 , tf , Ns , ord, fig) ; with your chosen initial vector y0 to create a solution matrix y1 . Make sure that your initial vector is on the attractor! Repeat with another nearby initial vector to create a second solution matrix y2 . Type dy = y2-y1 ; to create the diﬀerence. Either of the commands nn = sum(abs(dy)’ )’ ; or nn = sqrt(sum(dy’ .* dy’ ))’ ; will create a column vector of norms of the diﬀerences for all times. Then plot(t, log(nn)) ; will plot the logarithm of the norm versus time, and polyfit(t, log(nn), 1) ; will give the slope and intercept of the best linear ﬁt....
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