MATH103 April 2013
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[hide]Question 05 (a)
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Consider the iterated map . Calculate the steady states (fixed-points, equilibria) and mark them in the sketch below. Determine the stability of each equilibrium (explain analytically or graphically). Plot a cobweb starting with just above 2 (the point is indicated in the sketch).
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
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[show]Hint
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Recall in this setting, steady states are found by setting .
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[show]Solution
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The steady states are given by solving
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Subtracting x from both sides yields
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Using the quadratic formula gives
Analytically to check stability, we would need to take the derivative of our function and then plug in the fixed points and check that the absolute value of the derivatives in the fixed points is less than 1 (this would show that the fixed point is stable). It's easier to notice that graphically, none of the fixed points are the limit of an iterative sequence and so none of the fixed points are stable.
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