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119A-WQ05 - MAT119A Final Exam Write solutions in the exam...

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MAT119A Final Exam March 18, 2005 Write solutions in the exam booklet provided. Start each problem on a new page. Be sure to label diagrams carefully and justify all of your answers. Remember to print your name on the exam booklet. Good Luck! 1. Consider the equation dx dt = αx - x 1 + x , (1) with α > 0 and x > - 1. (a) Calculate the location of all fixed points x * of equation 1 as a function of the parameter α . (b) Determine the stability of these fixed points analytically . Use graphical means to determine stability of fixed points when the analytical method is inconclusive. (c) Plot the bifurcation diagram for the system using α as the control parameter. Indicate the stability of the fixed points on the diagram. At what values of x and α does a bifurcation occur? What type of bifurcation is it? (d) What type of bifurcations would occur if dx dt = αx - x 1 + x + β, β 6 = 0 . Briefly explain why? 2. Consider the nonlinear system of ODEs dx dt = x ( x - x 2 - y ) dy dt = y ( x - 1) (2) (a) Find all fixed points and use linear stabilty analysis to assess their stability. What conclusions
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