Bifurcations of the Logistic Equation




This applet demonstrates the changing shape of the logistic function f(x) = ax(1-x) as a varies. To begin select an initial value of a. Then select the number of compositions (k, for f^k). The applet will draw the graph of f^k and the diagonal line y=x. The value of a will change each step according to what is specified in the 'increment' box (this can be changed by the user).

Using this applet, one can see how the doubling bifurcations happen. For example, at parameter a=3, the fixed point of f^2 splits into three points. This indicates that as a increases through 3, a period two orbit of f appears. Note that the graph of f^2 at the fixed point has slope exactly 1 when a = 3; it goes from being a stable fixed point to being an unstable fixed point as the period two orbit appears. Similarly, by looking at f^4 one can observe the bifurcation of this period 2 orbit into a period 4 orbit orbit at a= 3.45. When a is between 3 and 3.45, f^4 has just 3 fixed points; one being the fixed point of f and the other two from the period 2 orbit. At a=3.45, the two fixed points of f^4 corresponding to the period two orbit split into three points; the 4 new fixed points are the period 4 orbit. Note that the stability of the period 2 orbit has changed from being stable to being unstable (the fixed point remains unstable).

One can also understand the 'self-similarity' of the final state diagram by considering various parts of f^k; see pages 625 and 633 of the text.