Bifurcation Diagrams

Configure parameter sweeps, retained points, Poincare sections, and Hopf examples to visualize qualitative changes in dynamics.

Steps

1. Open the bifurcation workflow in the toolbox.
2. Select the system to sweep. Start with Logistic for a fast map example or Lorenz for a flow example.
3. Choose the control parameter, for example r for Logistic or rho for Lorenz.
4. Set a coarse interval first, such as r = 2.8 to 4.0 for Logistic or rho = 0 to 60 for Lorenz.
5. Remove enough transient behavior so the plot shows retained long-term values rather than startup motion.
6. Increase the number of parameter samples and retained points only after the interval is meaningful.
7. Export a large figure when branches and dense regions are legible.

Theory

A bifurcation diagram is a compact visual summary of how a dynamical system changes when one parameter varies. For each parameter value, the simulation discards the transient and plots retained long-term values. A single branch suggests a stable fixed behavior; two, four, or many branches suggest periodic structure; dense bands suggest irregular or chaotic regimes.

For maps, the retained values are direct iterates. For flows, the toolbox needs an event or measurement rule, such as local maxima, final retained states, or a Poincare crossing. That rule matters because it determines which part of the continuous trajectory becomes a plotted point.

A Hopf bifurcation is a specific local mechanism in which an equilibrium changes stability and an oscillation is born or destroyed. It belongs in the same visual family of parameter-change diagrams, but it should not be confused with a full route to chaos.

How Parameters Change This Figure

• Parameter range: too wide can compress important branches; too narrow can hide the route into the behavior you want to show.
• Parameter samples: too few samples make the diagram look empty or jagged. Increase density for final plots.
• Transient length: too short leaves startup points in the chart. Increase burn-in before interpreting branches.
• Retained points: too few retained values hide dense bands; too many values can overplot into a solid block.
• Integration step: for flows, a large step can create numerical artifacts that look like bifurcations.
• Poincare plane: changing the section plane or crossing direction changes which returns are plotted.