Bifurcation diagram of the logistic map

The period doubling bifurcation of the logistic map can be visualized via a bifurcation diagram. Roughly speaking, a periodic point is called attracting if nearby trajectories converges towards it. It has been shown that there is at most one periodic attractor in the logistic map. Therefore we can show the periodic attractor by iterating the map from an initial point many times for the transient behavior to die down, and then plotting the subsequent trajectory. In the bifurcation diagram shown below, the horizontal axis denote the parameter a corresponding to the map, and the vertical axis plots the points of the trajectory (after the transients has die down).

We see that for 0 < a < 3, there is a periodic attractor consisting of one point (a fixed point), then at a=3, a period-doubling occurs where a period 2 attractor appears. As a is further increased, the period-2 attractor becomes unstable and a period 4 attractor occurs. This period doubling occurs infinitely many times as a is increased.

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