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Unformatted text preview: 1 CDS 140a: Homework Set 6 Due: Friday, November 20, 2009. 1. Let a be a real parameter with 0 a 4. The logistic map is the map of the unit interval [0 , 1] to itself that is defined by f ( x ) = ax (1- x ). Find the fixed points of f and determine their stability. 2. A two cycle of a map f is a point p together with its image q = f ( p ) with the property that f ( q ) = p . Show that the logistic map has a two cycle if a > 3. 3. The standard map is the map of the plane R 2 to itself that is given by x n +1 = x n + y n +1 y n +1 = y n + k sin x n , where k is a constant. Compute the Jacobian determinant of the associated map f and conclude that the standard map is area preserving. 4. Perform a stability analysis of the fixed point at the origin for the standard map. 5. Suppose that f : R 2 R 2 is an area preserving map and that the origin is a fixed point; that is, f (0 , 0) = (0 , 0). Is there a sense in which the eigenvalues of the linearization are symmetric in the unit circle? Verify this assertion forof the linearization are symmetric in the unit circle?...
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- Fall '09