AnswerPSet_2010_05

AnswerPSet_2010_05 - Answers P-Set Number 5, 18.385j/2.036j...

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Unformatted text preview: Answers P-Set Number 5, 18.385j/2.036j MIT (Fall 2010) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) November 16, 2010 Course TA: Jan Molacek, MIT, Math. Dept. room 2-331, Cambridge, MA 02139. Email: molacek@math.mit.edu Contents 1 Problem 07.05.05 - Strogatz (Find relaxation oscillation period). 2 1.1 Problem 07.05.05 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem 07.05.05 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Problem 07.06.18 - Strogatz (Mathieu eqn. and a super-slow time scale). 4 2.1 Problem 07.06.18 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Problem 07.06.18 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Problem 08.01.06 - Strogatz (Bifurcations in the phase plane). 6 3.1 Problem 08.01.06 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Problem 08.01.06 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Problem 08.02.07 - Strogatz (Hopf & homoclinic bifurcations via computer). 10 4.1 Problem 08.02.07 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Problem 08.02.07 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Problem 08.02.12 - Strogatz (Analytical criterion for Hopf bifurcations). 13 5.1 Problem 08.02.12 statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Problem 08.02.12 answer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 List of Figures 1.1 Problem 07.05.05. Relaxation oscillation limit cycle. . . . . . . . . . . . . . . . . 3 3.1 Problem 08.01.06. Nullclines and flow field. . . . . . . . . . . . . . . . . . . . . . 7 3.2 Problem 08.01.06. Nullclines and flow field. . . . . . . . . . . . . . . . . . . . . . 7 3.3 Problem 08.01.06. Far field phase plane portrait. . . . . . . . . . . . . . . . . . . 9 3.4 Problem 08.01.06. Phase plane portrait. . . . . . . . . . . . . . . . . . . . . . . . 10 4.1 Problem 08.02.07. Stable spiral point before a Hopf bifurcation. . . . . . . . . . . 11 4.2 Problem 08.02.07. Soft (supercritical) Hopf bifurcation. . . . . . . . . . . . . . . 11 4.3 Problem 08.02.07. Limit cycle about to undergo a homoclinic bifurcation. . . . . 12 1 2 1 Problem 07.05.05 - Strogatz (Find relaxation oscillation period). 1.1 Statement for problem 07.05.05. Consider the equation d 2 x dt + ( | x | - 1) dx dt + x = 0 . (1.1) Find the approximate period of the limit cycle for 1. Hint. This problem can be done using exactly the same set of tricks used to calculate the period for the van de Pol equation limit cycle in the relaxation oscillation regime....
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This note was uploaded on 02/28/2011 for the course MATH 18.385j taught by Professor R. during the Fall '10 term at MIT.

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AnswerPSet_2010_05 - Answers P-Set Number 5, 18.385j/2.036j...

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