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R Question 8 (Units 7 and 8) 20 marks In the following, parts (a), (b), (c) and (d) should be attempted by hand, using a calculator where necessary but without using a computer. Parts (e), (f) and (g) will require the use of a computer. Consider the two-dimensional map Xn+1 = f(xn), where for x = (x y), f(x) = (- - 7 - a [2] y : -ax+by - 43 and where a and b are constant parameters with b > 2a > 0. (a) Determine the Jacobian matrix for this map, and state whether there are values of a and b for which the map is area-preserving. (b) The map has a fixed point at (0,0). Show that this is the only fixed point when b<l+a, but that there are two additional fixed points when b > 1 + a, and locate these fixed points. (c) Show that the fixed point at (0,0) is an attractor for b< 1 + a but becomes a saddle when b > 1+ a. For the remainder of this question, let a = 0.2 and b=2.77. At these parameter values, this map has a strange attractor. (d) Classify all the fixed points found in part (b) for these values of a and b. [2] [3] a 131