Time for my second explorable on this blog. Some of us in the Los Angeles Rationality meetup are going through the Santa Fe Institute’s Introduction to Complexity course together, and one of the topics covered is the logistic map. I remember studying the stability behavior of maps like these before with a slight variation of a cobweb plot, so I went ahead and implemented one in javascript:

You can click and drag the red dot to change the initial value $x_0$ that’s fed into the logistic map, and you can click on the number to the right of the diagram and move horizontally to change the parameter $R$ between 1 and 4. The pale trajectory emitting from the red dot shows the path of successive applications of the logistic map $x_{n+1}=Rx_n(1-x_n)$, by alternating between updating the x and y components of a point. One starts on the x axis at the point $(x_0,0)$, and then moves vertically to the point $(x_0,x_1)$ on the vertical parabola, then moves horizontally to the point $(x_2,x_1)$ on the horizontal parabola, and so forth, bouncing back and forth between the two parabolas encoding the update rule $x_{n+1}=Rx_n(1-x_n)$. This explorable is pretty rudimentary, and it applies the map for a fixed 64 iterations, but this is enough for you to see the transition from the map having a stable fixed point to having attractors that are loops that the trajectory gets sucked into. I am displaying the trajectory with transparency so one can easily see when the trajectory settles into a stable orbit, as the orbit darkens up as lines pile up on each other.