of the logistic map
In the study of dynamical systems, the iterated
Figure one illustrates the behavior of the logistic map where xn = 0.1 and r = 2.9. To the left, the orbit for the first one-hundred iterations is plotted as a timeseries. To the right, a
In figure two where r = 3.2, the long-term behavior is a stable cycle of period two.
In figure three where r = 3.628, the orbit evenutally falls into a stable cycle of period six.
In figure four where r = 3.9, we discover an aperiodic orbit.
Figure five is an interactive timeseries and final-state diagram. Experiment with different seeds and r parameters to see their long-term behavior.
By concatenating each final-state diagram as we vary r over some interval within [0, 4], we discover an interesting, self-similar, bifurcation diagram. In the study of dynamical systems, a
To more easily explore the bifurcation diagram, you can zoom in on a region by clicking and dragging. Offset and limit parameters are introduced to help render regions at higher magnifications. Offset modifies how many iterations are not plotted, thereby allowing more time for convergence. Limit modifies the number of plotted iterations. The feathering that occurs near bifurcations can be refined by increasing the offset, and at high maginifcations, sparse regions can be populated more densely by increasing the limit parameter.