hw4(1) - Make sure to provide a print-out of the commands...

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Homework assignment 4 * March 14, 2006 1. Carry out the phase plane analysis for the Lotka-Volterra system: ˙ y 1 = r 1 y 1 [1 - y 1 - α 12 y 2 ] ˙ y 2 = r 2 y 2 [1 - y 2 - α 21 y 1 ] assuming that 0 < α 12 , α 21 < 1. Draw the nullclines, steady states and indicate the direction of the vector Feld in the di±erent parts of the state space. Based on linearization, discuss stability of all steady states. Show that all solutions in the Frst quadrant converge to some steady state and show that almost all solutions converge to a particular steady state in the interior of the Frst quadrant. ²or the latter solutions this implies that in the long run, the species will coexist. 2. In class we analyzed the Brusselator: ˙ x = 1 - ( β + 1) x + αx 2 y ˙ y = - αx 2 y + βx Using the software of your choice, perform simulations for this system assuming that we Fx a = 2, and let b range from 2 to 4 with increments of 0 . 5. Consider the same three to four initial conditions for all cases. Plot the solutions in each case in a single phase plane diagram.
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Unformatted text preview: Make sure to provide a print-out of the commands you used to generate the simulations and plots. [or your convenience, I have put a sample of a Mathematica notebook on the class webpage for a similar problem. To execute a command in Mathematica, you need to simultaneously press the shift and enter button.] 3. This problem considers the Van der Pol oscillator: x = x + y-x 3 / 3 y =-x Here ( x, y ) R 2 . Show that (0 , 0) is the only steady state, and that it is a repellor (lineariza-tion has two eigenvalues with positive real part). Show that the polytope { ( x, y ) R 2 | x [-3 , 3] , y [-6 , 6] , y x + 6 , y x-6 } is a trapping region. Conclude that the Van der Pol oscillator must have a periodic solution. * MAP 4484 / 5489; Instructor: Patrick De Leenheer. 1...
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