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Unformatted text preview: 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 2 1. Let V be a one dimensional potential whose derivative is given by V ( x ) = ( x- 1)( x- 2)( x- 3) and consider the system x = v v =- V ( x )- v (a) Show that solutions of this system for any initial conditions exist for all time. (b) For = 0 argue that the system has both homoclinic and periodic orbits. (c) What does Liapunov theory or La Salles invariance principle say about the fate of solutions with > 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) =- 1 1 . Consider a candidate Lyapunov function V ( x,y ) = x 2 + y 2 ....
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This note was uploaded on 02/23/2010 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.
- Fall '09