hw1 - 3* (Glendinning 5.6) Analyze the xed points of the...

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MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Homework I Due January 14, 2010. Questions with a star have a numerical/plotting component. 1 Analyze the phase line of the equation ˙ x = x ( 1 - log x ) . (Consider only the case x 0.) Now solve the equation exactly. Relate the two approaches. 2 Compute the matrix exponential e Λ t from its definition as an infinite series for the following matrices Λ : ± λ 1 0 0 λ 2 ² , ± λ 1 0 λ ² , ± ρ ω - ω ρ ² . Do the same for the matrix function log ( 1 + Λ t ) . Bonus: (2003), Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Rev. 45 , 3–49, doi:10.1137/S00361445024180.
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Unformatted text preview: 3* (Glendinning 5.6) Analyze the xed points of the system x = x 2-y-1 , y = ( x-2 ) y . Plot the trajectories. 4 Show that the origin is a xed point of the system x = ( x 2 + y 2 ) x 2 x 2 + y 2 ( 1-ax 2-ay 2 )-y x 2 + 2 y 2 , y = ( x 2 + y 2 ) y 2 x 2 + y 2 ( 1-ax 2-ay 2 )+ x x 2 + 2 y 2 , and discuss its nature. For a > 0 show that r > 0 for small r and r < 0 for large r . The Poincare Bendixson theorem shows that there is then a periodic orbit; explain why this makes sense from a graphical perspective. Now transform to polar coordinates, nd the periodic orbit and solve for r ( ) in the general case. 1...
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