MATH503 hw6 - 2. Consider the competing species system ˙ x...

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Homeworks (Math 503) T1 : Div, grad, curl and all that , by Schey (4th edition) T2 : Calculus of variations T3 : Nonlinear dynamics and chaos , by Strogatz Note: Detail your work to receive full credit Homework#6: (due Fri Dec 3 before noon) 1. For the following nonlinear systems (a) ˙ x = sin( y ) , ˙ y = x - x 3 (b) ˙ x = y + x - x 3 , ˙ y = - y - Find all the fixed points and study their linear stability. - Based on linear stability, sketch the trajectories near these fixed points and try to fill in the rest of the phase portrait.
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Unformatted text preview: 2. Consider the competing species system ˙ x = x ± 14-1 2 x-y ² , ˙ y = y ± 16-1 2 y-x ²- Find all the fixed points and study their linear stability.- Assuming x ( t ) ≥ 0 and y ( t ) ≥ 0, put the information together in a phase portrait. Discuss the long-term evolution of the two species based on these results. 3. Using the identities θ = tan-1 ± y x ² , r 2 = x 2 + y 2 show that ˙ θ = x ˙ y-y ˙ x r 2...
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This note was uploaded on 02/20/2011 for the course MATH 503 taught by Professor Schleiniger,g during the Fall '08 term at University of Delaware.

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