Logistic map bifurcation diagram

Parametric equations

The logistic map $x_{n+1}=r{x}_n(1-x_n)$ models a population with limited resources. As $r$ grows, the orbit goes from a fixed point through period 2, 4, 8, … and then into chaos. The bifurcation diagram puts $r$ on the horizontal axis and the long-run orbit values on the vertical axis, and it shows Feigenbaum universality.

Bifurcation diagram: for each $r$, drop the transient and plot the attractor set of $\{x_n\}$.

Interaction

• Parameter r: change $r$ and watch the orbit go from a fixed point through period doubling into chaos
• Initial value x₀: change the starting point, and compare the sensitivity to initial values in the chaotic region with the insensitivity in the periodic region
• Bifurcation / Orbit / Cobweb: switch among three readings and compare the long-run distribution, the time series, and step-by-step iteration
• Feigenbaum: mark the ratio of the gaps between period doublings and compare it with the universal constant

What to notice

• The bifurcation diagram projects the long-run attractors onto one parameter plane, so fixed points, period doubling, and chaotic bands are visible at once.
• The orbit and cobweb views stay at the current $r$. They explain one vertical slice of the bifurcation diagram through local dynamics.
• The Feigenbaum marks show how quickly the period-doubling gaps shrink, a scaling law at the edge of chaos.

Related

• Julia set
• Sierpinski triangle
• Vector field streamlines

Further reading

• Logistic map (Wikipedia)
• Bifurcation diagram (Wikipedia)