Calculator
Formula Used
The logistic map is a discrete nonlinear recurrence: xn+1 = r · xn · (1 − xn). Here x is a normalized population-like state, and r is the growth parameter.
A bifurcation dataset is built by sweeping r over a range, iterating the map many times, discarding early transient values, and keeping the last few states as representative long-run behavior for each r.
How to Use This Calculator
- Choose r minimum, r maximum, and a suitable r steps value.
- Set x₀ between 0 and 1 to start iterations.
- Use larger total iterations for cleaner long-run output.
- Increase discard to remove transient settling behavior.
- Set sample points to capture the final cycle or chaos band.
- Press Submit to preview results, then download CSV or PDF.
Example Data Table
| r min | r max | r steps | x₀ | iterations | discard | sample | Typical observation |
|---|---|---|---|---|---|---|---|
| 2.8 | 3.2 | 600 | 0.2 | 1200 | 400 | 60 | Single fixed point becomes a two-cycle. |
| 3.2 | 3.6 | 900 | 0.2 | 1600 | 600 | 80 | Period-doubling cascade with narrow stability windows. |
| 3.6 | 4.0 | 1200 | 0.2 | 2000 | 800 | 120 | Chaotic bands with occasional periodic islands. |
1) What the bifurcation output represents
This calculator generates a bifurcation dataset for the logistic map, a compact model of nonlinear feedback. For each growth value r, the map is iterated and the final states are recorded. Plotting x versus r reveals stability, cycles, and chaotic bands.
2) Parameter ranges used in practice
A common scientific sweep is r ∈ [2.5, 4.0]. In this interval the system transitions from a stable fixed point to periodic behavior and then to chaos. Many studies focus on r ≈ 3.0–4.0 where qualitative changes are densest and most informative.
3) Transient removal and sampling depth
Early iterations can reflect initial-condition settling rather than long-run dynamics. The discard setting removes these transients. The sample setting captures the last states after settling. In periodic regions, the sample reveals a small set of repeated values; in chaos, it fills a band.
4) Period-doubling and key numeric landmarks
As r increases, the logistic map undergoes period-doubling bifurcations: 1-cycle to 2-cycle, then 4, 8, and beyond. The onset of widespread chaos is near r ≈ 3.5699456 (the accumulation point). The scaling of bifurcation intervals is governed by the Feigenbaum constant δ ≈ 4.669, a hallmark of universality in nonlinear systems.
5) Reading stable windows inside chaos
Even in chaotic regimes, periodic “islands” appear. A well-known example is the period‑3 window near r ≈ 3.83. Increasing r steps improves resolution so these windows become visible in plots built from the exported dataset. Higher sampling highlights multi-band splitting and merging.
6) Sensitivity to initial conditions
For chaotic r, two close starting values x₀ diverge rapidly. This sensitivity is why repeated experiments often analyze ensembles rather than a single trajectory. You can test sensitivity by changing x₀ slightly and comparing the sampled values at the same r.
7) Quantifying chaos with the Lyapunov idea
A standard quantitative indicator is the Lyapunov exponent, which becomes positive when nearby trajectories separate exponentially. While this page focuses on bifurcation points, the exported (r, x) data can support downstream calculations and visualization workflows that estimate stability across the sweep.
8) Choosing settings for reliable datasets
For smooth results, keep r steps high (e.g., 800–2000) and use moderate sample values (50–150). Increase iterations and discard when exploring near transition points. The performance cap encourages balanced settings while still producing publication-ready CSV exports. Small parameter changes can reveal sharp regime boundaries in detail.
1) What is a bifurcation diagram in this context?
It is a plot of long‑run logistic map states x versus growth parameter r. Each vertical slice at a fixed r shows the attractor values after transients are removed.
2) Why do we discard early iterations?
Early steps can be dominated by the chosen x₀ rather than the attractor. Discarding iterations reduces bias and better reflects steady behavior, especially near bifurcations.
3) What do multiple points at one r mean?
Multiple values indicate a periodic cycle or chaotic band. Two values suggest period‑2, four values suggest period‑4, and a dense vertical cloud suggests chaos.
4) How should I pick r steps?
Use more steps for higher resolution. Around r ≈ 3.4–4.0, transitions are dense, so 1000+ steps often gives clearer structure, provided the point cap is respected.
5) Why does changing x₀ sometimes change results?
In stable periodic regions, different x₀ values usually converge to the same cycle. In chaos, nearby starts diverge, so sampled values may differ even for identical r.
6) My output says it was capped. What should I do?
Reduce r steps or sample so that steps × sample is smaller. You can also lower rounding or narrow the r range for a focused study.
7) How do I visualize the CSV?
Plot a scatter chart with r on the horizontal axis and x on the vertical axis. Use small markers and no connecting lines to reproduce the classic bifurcation structure.