Calculator Inputs
Example Data Table
Sample: r = 3.20, x0 = 0.40| Iteration | xn |
|---|---|
| 1 | 0.768000 |
| 2 | 0.570163 |
| 3 | 0.784247 |
| 4 | 0.541452 |
| 5 | 0.794502 |
| 6 | 0.522460 |
Formula Used
Core recurrence: xn+1 = r × xn × (1 − xn)
Lyapunov exponent estimate: λ ≈ (1 / m) Σ ln |r(1 − 2xn)| over the retained iterations.
Period detection: the calculator compares recent retained values against earlier values separated by candidate periods up to the chosen limit.
Threshold share: values above the chosen threshold are counted and divided by retained observations to show occupancy above that level.
How to Use This Calculator
- Select Single run for one parameter value or Parameter sweep for many r values.
- Enter the initial value x0, total iterations, and how many transient steps to discard.
- Choose a tolerance and a maximum period for orbit detection.
- Set a threshold if you want occupancy analysis above a reference level.
- Press the submit button to display the result directly below the header.
- Use the export buttons to download the retained results as CSV or PDF.
Frequently Asked Questions
1. What does the logistic map model?
It models nonlinear recursive growth where each next value depends on the current value and a control parameter. It is widely used to study stability, bifurcations, and chaos.
2. Why should I discard early iterations?
Early iterations often reflect transient behavior rather than the long-run attractor. Discarding them helps metrics focus on the steady or chaotic regime you actually want to analyze.
3. What does a positive Lyapunov exponent mean?
A positive Lyapunov exponent indicates sensitive dependence on initial conditions. Nearby starting values separate quickly, which is a hallmark of chaotic behavior in the map.
4. What does period detection tell me?
It estimates whether the sequence repeats after a fixed number of retained steps. A detected period of 1 suggests a fixed point, while larger periods indicate cyclical attractors.
5. Why must x0 stay between 0 and 1?
The classic logistic map is usually studied on the unit interval. Starting there makes the interpretation of bounded population-like dynamics consistent and easier to compare.
6. When should I use sweep mode?
Use sweep mode when you want to compare many parameter values quickly. It helps you locate stable zones, periodic windows, and regions where chaos becomes dominant.
7. What does the threshold analysis show?
It counts how often retained values stay above a chosen level. This is helpful when you want occupancy, persistence, or exceedance behavior instead of only average values.
8. Can this replace a full bifurcation diagram?
It gives strong numerical insight, especially in sweep mode, but a full bifurcation diagram still provides richer visual structure. This tool is best for fast calculation and reporting.