Bifurcation Diagram

What is a Bifurcation Diagram?

A bifurcation diagram shows how the long-run behaviour of a map \(f_r(x)\) changes as the parameter \(r\) is varied. For each value of \(r\):

• The map is iterated from \(x_0\) for a long transient (discarded).
• The subsequent iterates \(x_n\) are plotted as dots above that value of \(r\).
• A stable fixed point produces one dot; a period-2 orbit produces two; chaos produces a cloud.

Period-Doubling Cascade (Logistic Map)

For the logistic map \(f_r(x)=rx(1-x)\), increasing \(r\) from 1 to 4 reveals:

• \(r \in (1,3)\): unique stable fixed point \(x^*=(r-1)/r\).
• \(r \approx 3\): first bifurcation — fixed point loses stability, period-2 orbit born.
• \(r \approx 3.449\): period-4 born. Then period-8 at \(r\approx3.544\), period-16 at \(r\approx3.564\), \(\ldots\)
• \(r \approx 3.5699\) (Feigenbaum point \(r_\infty\)): onset of chaos.
• \(r=4\): fully chaotic; orbit is dense in \([0,1]\); Lyapunov exponent \(\lambda=\ln 2\).

The Feigenbaum Constant \(\delta\)

The bifurcation values \(r_1, r_2, r_3, \ldots\) at which the period doubles converge at a universal rate:

This constant \(\delta\) is universal — it appears in the period-doubling cascade of any smooth unimodal map, not just the logistic map. The tent and sine maps share the same \(\delta\).

Lyapunov Exponent Colouring

Dots are coloured by the Lyapunov exponent \(\lambda(r)=\lim_{N\to\infty}\frac{1}{N}\sum_{i=0}^{N-1}\ln|f'(x_i)|\):

• Blue (\(\lambda<0\)): stable — orbits converge. Stable fixed points and periodic windows.
• Amber (\(\lambda\approx 0\)): neutral — bifurcation boundary.
• Red (\(\lambda>0\)): chaotic — exponential divergence of nearby orbits.

The Lyapunov exponent is also plotted separately below the bifurcation diagram, showing exactly where chaos begins.

Periodic Windows Inside Chaos

Even within the chaotic regime (\(r>r_\infty\)) there are infinitely many periodic windows — narrow ranges of \(r\) where the orbit becomes periodic again. The most visible is the period-3 window near \(r\approx3.8284\). By Li–Yorke's theorem, period-3 implies all periods.

Other Maps

• Tent map \(f_\mu(x)=\mu\min(x,1-x)\): piecewise linear, same Feigenbaum constant, chaos at \(\mu=2\).
• Sine map \(f_r(x)=r\sin(\pi x)\): smooth, same universal behaviour, fully chaotic at \(r=1\).
• Quadratic \(f_c(x)=x^2+c\): Julia set parameter, chaos at \(c=-2\); related to Mandelbrot set on real axis.

How to Interact

• Scroll on either canvas to zoom the parameter axis.
• Drag to pan both axes simultaneously.
• Click on either canvas to set a parameter crosshair — the readout shows the exact value.
• Double-click to reset the view.
• Compute runs the full calculation; Animate sweeps \(r\) left to right.
• Toggle Feigenbaum lines to show/hide the annotated bifurcation points.