AnswerPSet_2010_02

AnswerPSet_2010_02 - Answers P-Set Number 2 18.385j/2.036j...

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Answers P-Set Number 2, 18.385j/2.036j MIT (Fall 2010) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) October 1, 2010 Course TA: Jan Molacek, MIT, Math. Dept. room 2-331, Cambridge, MA 02139. Email: [email protected] Contents 1 Problem 2.2.13 - Strogatz (Skydiving). 2 1.1 Problem 2.2.13 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem 2.2.13 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Problem 2.4.09 - Strogatz (Stat. Mech. and the critical slowing-down). 7 2.1 Problem 2.4.09 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Problem 2.4.09 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Problem 3.4.06 - Strogatz (Find and classify bifurcations). 8 3.1 Problem 3.4.06 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Problem 3.4.06 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Problem 3.4.15 - Strogatz (First-order phase transition). 10 4.1 Problem 3.4.15 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Problem 3.4.15 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 Problem 3.5.4R - (Bead on horizontal rotating wire). 13 5.1 Problem 3.5.4R statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Problem 3.5.4R answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Problem 3.5.06 - Strogatz (Model problem about singular limits). 18 6.1 Problem 3.5.06 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Problem 3.5.06 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7 Problem 4.1.08 - Strogatz (Potentials for vector fields on the circle). 21 7.1 Problem 4.1.08 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.2 Problem 4.1.08 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8 Problem 4.5.03 - Strogatz (Excitable systems). 22 8.1 Problem 4.5.03 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.2 Problem 4.5.03 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1
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2 9 Problem 5.2.11 - Strogatz (Single eigenspace matrix). 25 9.1 Problem 5.2.11 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 9.2 Problem 5.2.11 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 10 Problem 6.1.11 - Strogatz (Computer generated phase portrait). 26 10.1 Problem 6.1.11 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 10.2 Problem 6.1.11 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 11 Problem 6.2.02 - Strogatz (A trapped solution). 27 11.1 Problem 6.2.02 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 11.2 Problem 6.2.02 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 12 Problem 6.3.10 - Strogatz (Fixed point with inconclusive linearization). 28 12.1 Problem 6.3.10 statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 12.2 Problem 6.3.10 answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 List of Figures 1.1 Problem 2.2.13. Function for equation determining terminal velocity. . . . . . . . 6 2.1 Problem 2.4.09. Algebraic versus exponential decay at a critical slowing down. . . 8 3.1 Problem 3.4.06. Bifurcation diagram for dx/dt = r - (1 + x ) - 1 x . . . . . . . . . 9 4.1 Problem 3.4.15. Double well potential and critical value of a parameter. . . . . . 12 5.1 Problem 3.5.4R. Bifurcation diagram for a bead on a rotating wire. . . . . . . . . 17 6.1 Problem 3.5.06. Solution with transients due to small parameter in equation. . . 20 8.1 Problem 4.5.03. Fixed points for ˙ θ = μ + sin( θ ) . . . . . . . . . . . . . . . . . . . . 23 8.2 Problem 4.5.03. Plot of the solution of an excitable system. . . . . . . . . . . . . 24 9.1 Problem 5.2.11. Phase portraits for degenerate nodes. . . . . . . . . . . . . . . . 26 9.2 Problem 5.2.11. Vector fields for degenerate nodes. . . . . . . . . . . . . . . . . . 26 10.1 Problem 6.1.11. Phase plane portrait for the ”Parrot” system. . . . . . . . . . . . 27 12.1 Problem 6.3.10. Nonlinear saddle point. . . . . . . . . . . . . . . . . . . . . . . . 31 1 Problem 2.2.13 - Strogatz (Skydiving). 1.1 Statement for problem 2.2.13. The velocity v = v ( t ) of a skydiver falling to the ground is governed by the equation m dv dt = mg - kv 2 , (1.1) where m is the mass of the skydiver, g is the acceleration due to gravity, and k > 0 is a constant related to the amount of air resistance (the drag constant ).
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3 (a) Obtain the analytical solution for v ( t ), assuming that v (0) = 0. (b) Find the limit of v ( t ) as t → ∞ . This limiting velocity is called the terminal velocity . (c) Give a graphical analysis of this problem, and thereby re-derive a formula for the terminal velocity.
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