Dynamical systems - Exercise 7
نویسنده
چکیده
1. Compute the center manifold (to second order), and the reduced system on it (to third order), and conclude regarding the behavior of the systems near the origin (draw phase portrait) for F(x, y) = (x + xy − y 2 , 1 2 y + x 2). Solution. Let us write the map in the form: x n+1 = x n + x n y n − y 2 n y n+1 = 1 2 y n + x 2 n (1) Note that the origin (0, 0) is the fixed point. Calculate the Jacobian in the origin DF (0,0) = 1 + y x + 2y 2x 1 2 (0,0) = 1 0 0 1 2 So λ 1 = 1 and λ 2 = 1 2. So we have E DF = E s ∪ E c. Where E s is the eigenspace corresponding to λ 2 = 1 2. And E c corresponds to λ 1 = 1. Then ∃!W s which is tangent to E s at (0, 0). Let us seek the central manifold W c in the form W c = {(x, y)|y = h(x)}. (2) Assume that the discrete system is in the form x n+1 = Bx n + G(x n , y n) y n+1 = Cy n + H(x n , y n) (3) Substitute (2) into (3) and obtain y n+1 = h(x n+1) = h(Bx n + G(x n , y n)) = Cy n + H(x n , y n), or N (h(x)) = h(Bx n + G(x n , y n)) −Cy n − H(x n , y n) = 0 (4) We need to compute the center manifold to the second order, hence let y = h(x) = ax 2 + O(3).
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