hw1cds140b

hw1cds140b - 4. Consider a gradient system. Show that, away...

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1 CDS 140b: Homework Set 1 Due by Tuesday, January 20, 2009. 1. Determine whether the following system can have any periodic orbits: ˙ x = y 2 + y cos x and ˙ y = 2 xy + sin x. 2. Use index theory to show that the system ˙ x = x (4 - y - x 2 ), ˙ y = y ( x - 1) cannot have any periodic orbits. 3. Show that the system ˙ x = y - x 3 , ˙ y = - x - y 3 cannot have any periodic orbits by considering a Liapunov function V = ax 2 + by 2 with suitable a,b . Can you think of another way ( i.e. without Liapunov functions) of proving this statement?
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Unformatted text preview: 4. Consider a gradient system. Show that, away from fixed points, the trajectories of the system cross the level sets of V at right angles, and that V is decreasing along the trajectories. 5. Consider the given by ¨ x + x = μ (1-x 2 ) ˙ x with μ > 0. Use Bendixson’s criterion to show that periodic solutions necessarily have to cross x = ± 1....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.

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