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Unformatted text preview: Answers P-Set Number 3, 18.385j/2.036j MIT (Fall 2010) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) October 19, 2010 Course TA: Jan Molacek, MIT, Math. Dept. room 2-331, Cambridge, MA 02139. Email: [email protected] Contents 1 Problem 5.1.10 - Strogatz (Attracting and Liapunov stable). 2 1.1 Problem 5.1.10 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem 5.1.10 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Problem 6.1.10 - Strogatz (Computer generated phase portrait). 3 2.1 Problem 6.1.10 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Problem 6.1.10 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Problem 6.3.17 - Strogatz (Nasty fixed point). 6 3.1 Problem 6.3.17 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Problem 6.3.17 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Problem 6.5.19 - Strogatz (Rabbits versus foxes). 9 4.1 Problem 6.5.19 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Problem 6.5.19 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Problem 07.01.09 - Strogatz (Circular pursuit problem). 11 5.1 Problem 07.01.09 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Problem 07.01.09 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 Problem 07.02.16 - Strogatz (Dulac’s criterion needs simply connected). 16 6.1 Problem 07.02.16 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 Problem 07.02.16 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7 Problem 07.02.17 - Strogatz (Dulac’s criterion on an annulus). 17 7.1 Problem 07.02.17 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7.2 Problem 07.02.17 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 List of Figures 2.1 Problem 6.1.10. Phase plane portrait for the ”two-eyed monster”. . . . . . . . . . 4 2.2 Problem 6.1.10. Phase plane portrait for the ”two-eyed monster”. . . . . . . . . . 5 1 2 2.3 Problem 6.1.10. Phase plane portrait for the ”two-eyed monster”. . . . . . . . . . 6 3.1 Problem 6.3.17. Phase plane portrait for a nasty fixed point. . . . . . . . . . . . 8 5.1 Problem 07.01.09. Dog-Duck coordinate system. . . . . . . . . . . . . . . . . . . 11 5.2 Problem 07.01.09. Dog–duck system for k = 1 . 0. . . . . . . . . . . . . . . . . . . 14 5.3 Problem 07.01.09. Dog–duck system for k = 0 . 5. . . . . . . . . . . . . . . . . . . 15 7.1 Problem 07.02.17. Limit cycles in an annular region. . . . . . . . . . . . . . . . . 18 1 Problem 5.1.10 - Strogatz (Attracting and Liapunov stable)....
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