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Final2008

# Final2008 - 1 CDS 140a Final Examination J Marsden December...

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1 CDS 140a Final Examination J. Marsden, December, 2008 Attempt SEVEN of the following ten questions. Each question is worth 20 points. The exam time limit is three hours; no aids are permitted . Turn in the exam by 5pm on Thursday, December 11, 2008 . Print Your Name : The SEVEN questions to be graded : 1 2 3 4 5 6 7 8 9 10 /140

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2 1. Let V be a one dimensional potential whose derivative is given by V 0 ( x ) = ( x - 1)( x - 2)( x - 3) and consider the system ˙ x = v ˙ v = - V 0 ( x ) - νv (a) Show that solutions of this system for any initial conditions exist for all time. (b) For ν = 0 argue that the system has both homoclinic and periodic orbits. (c) What does Liapunov theory or La Salle’s invariance principle say about the fate of solutions with ν > 0? 2. Consider the following system in R 2 : ˙ x = x + y 2 ˙ y = - y. (a) Determine the stable and unstable subspaces of the linearization at the origin. (b) Calculate the explicit solution ( x ( t ) , y ( t )) of this system. (c) Find an explicit expression for the stable and unstable manifolds of this system. 3. Consider a system of the form ˙ x = f 1 ( x, y ) ˙ y = f 2 ( x, y ) where f 1 (0 , 0) = f 2 (0 , 0) = 0 and Df (0) = 0 - 1 1 0 . Consider a candidate Lyapunov function V ( x, y ) = x 2 + y 2 . (a) Assume that f 1 ( x, y ) and f 2 ( x, y ) are polynomials in x and y of degree at most two. Write ˙ V
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Final2008 - 1 CDS 140a Final Examination J Marsden December...

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