hw7 - 6 = 0 , 1 2 , 2. (b) (5 pts) What is G 2 (the...

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Applied Dynamical Systems - ME215A Fall 2010 Homework 7 - Due Wednesday November 17 1. (25 pts) Consider the vector field ˙ x = y + a 1 x 2 + b 1 xy + c 1 y 2 , (1) ˙ y = a 2 x 2 + b 2 xy + c 2 y 2 , (2) whose Jacobian evaluated at the fixed point ( x, y ) = (0 , 0) is ± 0 1 0 0 ! . Find the near-identity coordinate transformation x = ξ + α 1 ξ 2 + β 1 ξη + γ 1 η 2 , (3) y = η + α 2 ξ 2 + β 2 ξη + γ 2 η 2 (4) which gives a system of the form ˙ ξ = η + O (3) , (5) ˙ η = 2 + Bξη + O (3) . (6) That is, in terms of a i , b i , and c i , i = 1 , 2, what should you take for α i , β i , γ i , i = 1 , 2, and what are A and B ? 2. (25 pts total) Consider ± ˙ x ˙ y ! = ± 1 0 0 λ ! | {z } J ± x y ! (7) (a) (10 pts) Calculate R 2 = L (2) J ( H 2 ), where H 2 is the space of second order homogeneous polynomials. Verify that R 2 = H 2 if and only if
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Unformatted text preview: 6 = 0 , 1 2 , 2. (b) (5 pts) What is G 2 (the complement of R 2 ) when = 0? Use this to write down the form that the normal form must take, accurate to second order in x and y , for this case. (c) (5 pts) What is G 2 when = 1 2 ? Use this to write down the form that the normal form must take, accurate to second order in x and y , for this case. (d) (5 pts) What is G 2 when = 2? Use this to write down the form that the normal form must take, accurate to second order in x and y , for this case. 1...
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