12. (a) The fixed points are the solutions of xn = s*xn*(1 - xn2), that is, xn = 0, (s - 1)1/2/s, and -(s - 1)1/2/s. Only the first two lie in the range 0 <= x <= 1. Note the second and third fixed points are real only for s >= 1.

(b) The slope of the tangent line is s*(1 - 3xn2).

(c) At xn = 0, the slope of the tangent line is s; at xn = 1, the slope of the tangent line is -2s.

(d) The maximum occurs where the slope of the tangent line is zero, that is, at xn = 3-1/2.

(e) For xn = 3-1/2, xn+1 = 2s/33/2. Thus xn+1 = 1 for s = 33/2/2 and so the dynamics are bounded between 0 and 1 for 0 <= s <= 33/2/2.

(f) The fixed point xn = 0 is stable for 0 <= s <= 1. The slope of the tangent line at xn = (sqrt(s - 1))/s is 3 - 2*s, so this fixed point is stable for |3 - 2*s| < 1. That is, for 1 < s < 2.

Return to Exercises