The bifurcation diagram
A bifurcation diagram places a control parameter on one axis and the long-term states of an iterated system on the other. One picture shows stability splitting into complexity.
Building the diagram
For each parameter r in the logistic map, iterate long enough to discard the transient and plot later x values. A stable fixed point creates one branch; a period-two orbit creates two, then four, eight and increasingly dense bands.
For each parameter r, iterate x↦rx(1−x), discard an initial transient, and plot many subsequent states vertically above r. A stable fixed point gives one line, a period-two orbit gives two, and chaos produces a dense band. The number of plotted states controls visible density and should not be confused with transient length, which removes earlier behavior.
Universality in the route to chaos
The intervals between period doublings shrink toward a constant ratio associated with Feigenbaum. Similar scaling occurs across broad classes of nonlinear systems, making the diagram more than a portrait of one equation.
The spacing between successive period doublings in many unimodal maps shrinks according to a universal ratio approaching the Feigenbaum constant δ. Universality means shared asymptotic scaling under suitable conditions, not identical diagrams. A finite pixel grid resolves only a few bifurcations accurately. A demonstration should therefore include computed parameter values rather than reading the constant from a screenshot.
Chaos contains windows
Dense chaotic bands are interrupted by periodic windows, including a conspicuous period-three region. Fine magnification reveals smaller copies of the broad branching story, but finite sampling can miss narrow windows or create misleading gaps.
Periodic windows open inside the chaotic region, including a prominent period-three window. A fresh cascade of doublings begins inside it. These islands show why “larger parameter means more chaos” is too crude. Horizontal magnification without a longer transient can fill the windows with temporary states; zoom and observation time must increase together.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Simple mathematical models with very complicated dynamicsNature
- Lyapunov ExponentWolfram MathWorld
Continue the idea
Chaos is deterministic
Why deterministic equations can become unpredictable, how sensitivity differs from randomness and where fractals enter phase space.
Lyapunov fractals
How alternating parameter sequences turn the Lyapunov exponent of a logistic system into feathered stability maps.
Strange attractors
Why some dissipative chaotic systems settle onto bounded sets with sensitive motion and fractal geometry.