Calculator Inputs
Formula Used
Main recurrence: xn+1 = r xn(1 - xn)
Fixed points: x* = 0 and x* = 1 - 1/r, when r ≠ 0
Lyapunov estimate: λ ≈ (1/N) Σ ln |r(1 - 2xn)|
Interpretation: Negative λ suggests stable behavior. Positive λ suggests sensitive, chaotic behavior.
The logistic map is a classic nonlinear recurrence. It starts simple, yet it produces fixed points, cycles, bifurcations, and chaos. This calculator iterates the map, estimates long-run behavior, and compares nearby seeds to reveal sensitivity.
How to Use This Calculator
- Enter the growth parameter r.
- Enter the initial value x0.
- Choose how many iterations you want.
- Set a burn-in count to ignore early transients.
- Select display precision and attractor tail size.
- Enable the second seed to test sensitivity.
- Press Calculate Sequence.
- Review the summary cards, graphs, and full table.
- Export the results as CSV or PDF when needed.
Example Data Table
| r | x0 | Iterations | Burn-in | Typical observation |
|---|---|---|---|---|
| 2.50 | 0.20 | 20 | 5 | Sequence settles near a stable fixed point. |
| 3.20 | 0.20 | 40 | 10 | Sequence usually enters a period-2 cycle. |
| 3.50 | 0.20 | 60 | 20 | Period doubling becomes easy to observe. |
| 3.90 | 0.20 | 120 | 40 | Chaotic behavior and seed sensitivity become obvious. |
Frequently Asked Questions
1) What does the parameter r control?
It controls the growth strength of the recurrence. Small values usually damp the sequence. Larger values create oscillations, then period doubling, and finally chaos within the classic range.
2) Why should x0 usually stay between 0 and 1?
That interval matches the standard logistic map setup. Within it, the recurrence shows the best-known stability, cycle, and chaos patterns. Other values can still be computed, but interpretation changes.
3) What is burn-in?
Burn-in removes early transient terms from the statistics. This helps you study long-run behavior instead of initial settling. The full table still keeps every iteration for inspection.
4) What does the Lyapunov estimate mean?
It measures average stretching between nearby trajectories. Negative values usually indicate stable behavior. Positive values suggest strong sensitivity to starting conditions and chaotic dynamics.
5) Why compare a second seed?
A second seed helps you see sensitivity directly. Two nearly identical starts can remain close in stable regions, yet separate quickly in chaotic regions. The difference column shows that change numerically.
6) What are fixed points in this map?
Fixed points are values that map to themselves after one step. For this recurrence, they are 0 and 1 - 1/r, when r is not zero. Stability depends on the derivative around them.
7) Why does the graph sometimes look irregular?
Irregular plots often appear in transition or chaotic regions. The sequence may never settle to one value or a short cycle. That irregularity is a key feature of nonlinear dynamics.
8) What do CSV and PDF exports include?
The exports include the computed summary and the iteration table. This makes it easier to archive runs, compare parameters, or share results with students, colleagues, or clients.