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# hw1 - That is ﬁnd a and b in the equation E t ≈ a bt...

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Applied Dynamical Systems - ME215A Fall 2010 Homework #1 - Due Wednesday Oct 6th, in class 1. (20 pts total) Consider the linear set of ordinary differential equations for x = ( x, y ): ˙ x = A x , (1) where A = - 1 100 0 - 10 . (2) (a) (2 pts) Show that A is a non-normal matrix, i.e. AA T = A T A . (b) (4 pts) Find the eigenvalues and eigenvectors of A . Are the eigenvectors orthogonal? (c) (8 pts) Find the exact solution for x ( t ) in terms of the initial conditions x (0) = ( x 0 , y 0 ). You can do this by (i) going to new coordinates which diagonalize A , solving the equations in these coordinates, then transforming back to the original coordinates, or (ii) by solving the ˙ y equation for y , plugging this into the ˙ x equation and solving for x by finding a solution to the homogeneous equation and a particular solution. (d) (4 pts) Let E = x 2 + y 2 . Taylor expand E ( t ) about t = 0 up to linear order in time.
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Unformatted text preview: That is, ﬁnd a and b in the equation E ( t ) ≈ a + bt valid for small t . (e) (2 pts) Find one initial condition for which dE dt ² ² ² t =0 > 0, and one initial condition for which dE dt ² ² ² t =0 < 0. 2. (10 pts total) (a) (5 pts) Find the eigenvalues of a real 2 × 2 matrix A in terms of its trace and its determinant. Hence, ﬁnd the regions in the (Tr A, det A ) plane in which A has two negative real eigenvalues, two positive real eigenvalues, a pair of complex conjugate eigenvalues with negative real parts, etc. Also, ﬁnd the curve on which A has multiple eigenvalues. (b) (5 pts) For the planar diﬀerential equation ˙ x = Ax , x ∈ R 2 , sketch the phase portrait for each of the regions identiﬁed in (a). 1...
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