WinslowQuiz2Key

WinslowQuiz2Key - 1 Consider the map xn 1 = 2xn(1 xn Derive...

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1. Consider the map x n+1 = 2x n /(1+x n ) . Derive all fixed points and evaluate their stability. Show your work. Answer : 0 ) 1 ( 1 2 = + = x x x x x simplifies to… Therefore, x = 0 or 1 are the fixed points. Evaluate the first derivative to determine stability, by the quotient rule the derivative is: () 1 , 0 2 1 1 , 0 ) 1 ( * 2 ) 1 ( * 2 = = + + = x x x x x dx x df @ x = 0 , 2 = dx x df @ x = 1 , 5 . 0 = dx x df Therefore, the fixed point at 0 is unstable and the fixed point at 1 is stable. 2 . For each map below, generate cobweb plots to determine the map’s behavior (ignore ranges of inputs that give you imaginary numbers. Use these plots in order to fully describe all fixed points and their stability. a. x n+1 =sqrt(x n ) Answer
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This map is restricted to the positive quadrant and has fixed points at 0 and 1 (determined by where the function crosses the line of slope 1). All orbits started near 0 end up at 1 and more generally it can be noted that all orbits not started at 0 will end up at 1. In the figure above 2 example orbits are represented, one starting at 0.01 and one at 4 (the star
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This note was uploaded on 05/27/2008 for the course BME 223 taught by Professor Winslow during the Spring '08 term at Johns Hopkins.

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WinslowQuiz2Key - 1 Consider the map xn 1 = 2xn(1 xn Derive...

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