Visualizing bifurcations

of the logistic map

In the study of dynamical systems, the iterated logistic map is a canonical example of a simple, deterministic function which exhibits a surprising array of behavior: stable fixed points, periodic orbits, aperiodic orbits, etc. Mathematically, the logistic map is written as xn+1 = rxn(1 − xn) where xn is a number in the interval [0, 1] and the parameter r is in the interval [0, 4].

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 final-state diagram summarizes the long-term behavior: a single, stable fixed point near 0.655.

Fig. 1: Time series and final state diagram for x0 = 0.1 and r = 2.9

In figure two where r = 3.2, the long-term behavior is a stable cycle of period two.

Fig. 2: Time series and final state diagram for x0 = 0.1 and r = 3.2

In figure three where r = 3.628, the orbit evenutally falls into a stable cycle of period six.

Fig. 3: Time series and final state diagram for x0 = 0.1 and r = 3.628

In figure four where r = 3.9, we discover an aperiodic orbit.

Fig. 4: Time series and final state diagram for x0 = 0.1 and r = 3.9

Figure five is an interactive timeseries and final-state diagram. Experiment with different seeds and r parameters to see their long-term behavior.

Fig. 5: Interactive time series plot and final state diagram for logistic map

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 bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system (r in the case of the iterated logistic map).

Fig. 6: Interactive bifurcation diagram of the logistic map

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.