Calculator Inputs
Plotly Graph
The chart plots steady-state values of the logistic map against the control parameter.
Submit the form to build the bifurcation diagram and unlock CSV and PDF export.
Formula Used
Logistic map recurrence:
xn+1 = r xn (1 - xn)
Here, r is the control parameter and xn is the state at iteration n. For each selected parameter value, the calculator first removes transient behavior, then records later iterations. Those retained points form the bifurcation diagram.
A rough Lyapunov estimate is also reported for the inspection parameter:
λ ≈ (1/N) Σ ln |r(1 - 2xn)|.
Positive values usually indicate sensitive dependence and chaotic behavior, while negative values suggest convergence toward stable periodic structure.
How to Use This Calculator
- Enter the parameter start and end values for the diagram.
- Choose how many parameter steps should be sampled.
- Set the initial value, transient iterations, and recorded iterations.
- Pick an inspection parameter to analyze local orbit behavior.
- Adjust decimal precision and point size for readability.
- Press Generate Diagram to show the result under the header.
- Review the summary cards, preview table, and Plotly chart.
- Use the export buttons to save the generated data as CSV or PDF.
Example Data Table
| Parameter r | Typical behavior | Representative steady values |
|---|---|---|
| 2.8000 | Stable fixed point | 0.6429 |
| 3.2000 | Period-2 orbit | 0.5130, 0.7995 |
| 3.5000 | Period-4 orbit | 0.3828, 0.8269, 0.5009, 0.8750 |
| 3.8300 | Periodic window | Several repeated attractor values |
| 3.9500 | Strongly chaotic region | Many nonrepeating steady-state points |
Frequently Asked Questions
1. What does a bifurcation diagram show?
It shows how the long-term behavior of an iterative system changes as a control parameter varies. Stable fixed points, periodic cycles, chaotic bands, and periodic windows all become visible in one graph.
2. Which equation does this calculator use?
This calculator uses the logistic map, xn+1 = r xn (1 - xn). It is a classic nonlinear recurrence used to study stability, period doubling, and chaos.
3. Why are transient iterations removed?
Early iterations often reflect starting conditions rather than steady behavior. Removing them helps the graph display the attractor structure that remains after the system settles into its long-term pattern.
4. What is the inspection parameter used for?
It selects one specific parameter value for closer analysis. The calculator estimates the orbit type there and reports a Lyapunov value to help identify stable, periodic, or chaotic motion.
5. What does a positive Lyapunov estimate mean?
A positive value usually suggests nearby starting points diverge over time, which is a hallmark of chaos. Negative values usually indicate attraction toward stable fixed points or periodic cycles.
6. Why do some parameter ranges look denser than others?
Chaotic regions produce many distinct steady values, so they appear thicker or denser. Periodic regions generate only a few repeating values, so they appear as separated branches or bands.
7. How should I choose parameter steps and recorded iterations?
Use more parameter steps for finer horizontal detail and more recorded iterations for richer vertical structure. Very high settings improve detail but increase processing time and file size.
8. Can I export the generated result?
Yes. The calculator lets you download the plotted data as CSV and the report as PDF after you generate a diagram. This helps with archiving, reporting, or further analysis.