x_{k+1} = Q(a, x_{k}) = a x_{k}(1 - x_{k}), k = 0, 1, 2, ...The initial guess x_{0} and the parameter a satisfy the conditions
0 <= x_{0} <= 1, and 0 <= a <= 4For values in these ranges the iterations remain bounded and satisfy 0 <= x_{n} <= 1 for all n. We are interested in the long term behavior of the sequence x_{0}, x_{1}, x_{2}, ..., x_{k}, ... for given values of a and x_{0}.
The applet also graphically shows the iterations using a staricase and cobweb diagram. The graph shows the parabola y = ax(1 - x) and the line y=x. The sequence of iterations is shown as a sequence of lines (vertically to the parabola horizontally to the line y=x, vertically to the parabola, horizontally to the line y=x, and so on). The x values of the vertical lines correspond to the terms in the sequence. For monotone convergence to a limit we have a staircase diagram (try a=1.9), for oscillating convergence to a limit we have a cobweb diagram (try a=2.9). For values of a > 3.0 there is no convergence. Either a periodic point is obtained or chaos is obtained. Try various values of a and analyze the behavior of the logistic equation. For the long term behavior it is best to skip a lot of iterations before plotting them. The actual values of the last few iterations are shown.