Logistic Map

The Logistic map is a second-order polynomial map with dynamics that ranges from deterministic to periodic to completely chaotic [¹].

The map is given by the equation \begin{align} \begin{split} x_{n+1} = r x_n (1-x_n), \ x_n\in[0,1] \end{split} \end{align} where parameter \(r\in[0,4]\). For \(r\in[0,1)\), the map has a single stationary point \(0\) and \(r=1\) is a so called bifurcation point after which a new attraction point arises. For \(r\in(1,3)\), the attraction point is \(\frac{r-1}{r}\). For \(r\in(1,3)\), the map dynamics slowly transitions from periodic to chaotic. For \(r=4\), the logistic map generates chaos - regardless of the initial value \(x_0\), that is, \(x_n\) covers the interval \([0,1)\) uniformly.

The following plot shows the bifurcation diagram of the logistic map for \(r\in[2.8,4]\). To zoom into the diagram, click and drag to select a rectangle. At any point you can return to the initial view with right click of the mouse.

References