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hw5 - r = 1 What is its period T(b(10 pts Find that exact...

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Applied Dynamical Systems - ME215A Fall 2010 Homework #5 - Due Wednesday November 3rd in class 1. (20 pts total) Consider the system ˙ x = x + y - x 3 - 6 xy 2 (1) ˙ y = - x 2 + 2 y - 8 y 3 - x 2 y (2) (a) (5 pts) Show that ( x, y ) = (0 , 0) is an unstable fixed point. (b) (10 pts) Let V = x 2 + 2 y 2 . Show that ˙ V 0 on the curve V = 1, and ˙ V 0 on the curve V = 1 / 2. (c) (5 pts) Given that there are no other fixed points besides the one at ( x, y ) = (0 , 0), what does the result in (b) imply about the existence of periodic orbits? 2. (25 pts total) Consider the system ˙ r = μr (1 - r ) , (3) ˙ θ = ω, (4) where ( r, θ ) are standard polar coordinates.
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Unformatted text preview: r = 1. What is its period T ? (b) (10 pts) Find that exact Poincar´ e map r n +1 = P ( r n ) using the Poincar´ e section Σ at θ = 0. (c) (5 pts) By evaluating the derivative of P at the periodic orbit, determine, in terms of μ , when the periodic orbit is stable and unstable. (d) (5 pts) Linearize (3) about r = 1 by setting r = 1 + y . Solve the resulting equation for y ( t ). How is y ( T ) related to y (0)? How is this related to the results of (c)? 1...
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