AnswerPSet_2010_04

AnswerPSet_2010_04 - Answers P-Set Number 4 18.385j/2.036j...

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Answers P-Set Number 4, 18.385j/2.036j MIT (Fall 2010) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) November 5, 2010 Course TA: Jan Molacek, MIT, Math. Dept. room 2-331, Cambridge, MA 02139. Email: [email protected] Contents 1 Problem 6.1.13 - Strogatz (Draw a phase portrait). 2 1.1 Problem 6.1.13 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem 6.1.13 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Problem 6.5.06 - Strogatz (Epidemic model revisited). 2 2.1 Problem 6.5.06 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Problem 6.5.06 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Problem 07.02.07 - Strogatz (A system both potential and Hamiltonian). 5 3.1 Problem 07.02.07 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Problem 07.02.07 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Problem 07.02.x1 - 385 xtra (Area evolution). 6 4.1 Problem 07.02.x1 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Problem 07.02.x1 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Problem 07.03.11 - Strogatz (Cycle graphs). 9 5.1 Problem 07.03.11 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.2 Problem 07.03.11 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Problem 07.05.06 - Strogatz (Biased van der Pol). 12 6.1 Problem 07.05.06 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Problem 07.05.06 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7 Problem 07.06.02 - Strogatz (Calibrating regular perturbation theory). 21 7.1 Problem 07.06.02 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.2 Problem 07.06.02 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8 Problem 07.06.14 - Strogatz (Computer test of two timing). 22 8.1 Problem 07.06.14 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.2 Problem 07.06.14 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1
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2 List of Figures 1.1 Problem 6.1.13. Phase plane portrait: 3 closed orbits and single fixed point. . . . 3 2.1 Problem 6.5.06. Kerrmack-McKendrick epidemic model. . . . . . . . . . . . . . . 5 3.1 Problem 07.02.07. A gradient system. . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1 Problem 07.03.11. Globally attracting cycle graph. . . . . . . . . . . . . . . . . . 10 5.2 Problem 07.03.11. Behavior as t → ∞ near a cycle graph. . . . . . . . . . . . . . 12 6.1 Problem 07.05.06. Biased van der Pol ( a 2 > 1.) Nullclines and flow field. . . . . . 14 6.2 Problem 07.05.06. Biased van der Pol ( a 2 < 1.) Nullclines and flow field. . . . . . 15 6.3 Problem 07.05.06. Biased van der Pol ( μ 1); phase portrait. . . . . . . . . . . 18 6.4 Problem 07.05.06. Biased van der Pol (0 < 1 - a 1); limit cycles. . . . . . . . . 20 8.1 Problem 07.06.14. Comparison between asymptotic and numerical solution. . . . 24 8.2 Problem 07.06.14. Error in asymptotic solution. . . . . . . . . . . . . . . . . . . . 25 1 Problem 6.1.13 - Strogatz (Draw a phase portrait). 1.1 Statement for problem 6.1.13. Draw a phase portrait that has exactly three closed orbits and one fixed point. 1.2 Answer for problem 6.1.13. From index theory we know that the orbits must be one inside the other, with the fixed point inside them all. A simple example of this can be constructed using polar coordinates (see figure 1.1 ): ˙ θ = 1 and ˙ r = 0 . 01 r ( r 2 - 1)( r 2 - 4)( r 2 - 9) . It is clear that this system has three closed orbits — the limit cycles at r = 1 (unstable), r = 2 (stable) and r = 3 (unstable) – and only one fixed point (the origin, a stable spiral point). 2 Problem 6.5.06 - Strogatz (Epidemic model revisited). 2.1 Statement for problem 6.5.06. (Epidemic model revisited).) In Exercise 3.7.6, you analyzed the Kerrmack-McKendrick model of an epidemic by reducing it to a first-order system. In this problem you will see how much easier the analysis becomes in the phase-plane. As before, let x ( t ) 0 denote the size of the healthy population and y ( t ) 0 denote the size of the sick population. Then the model is dx dt = - k x y, and dy dt = k x y - ‘ y, (2.1)
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3 -4 -2 0 2 4 -2 0 2 4 x y Three limit cycles In cartesian coordinates, the system is ˙ x = - y + 0 . 01 g ( r ) x , ˙ y = x + 0 . 01 g ( r ) y , where g = ( r 2 - 1)( r 2 - 4)( r 2 - 9) .
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