Bifurcation Diagram Generator

Formula Used

Logistic map: x_n+1 = r x_n(1 − x_n), where 0 < x < 1 and r controls growth and nonlinearity.

Tent map: x_n+1 = r x_n for x_n < 0.5, otherwise x_n+1 = r(1 − x_n). This piecewise rule creates sharp stretching and folding.

Bifurcation diagram: for each r, iterate the map, discard burn-in steps, then plot the remaining long-run samples (r, x).

How to Use This Calculator

1. Pick a map type and set an r-min and r-max interval.
2. Increase r-steps for smoother structure and finer windows.
3. Set iterations high enough to settle long-run behavior.
4. Use burn-in to remove transient dependence on x0.
5. Choose sample points to capture cycles and chaotic bands.
6. Click Generate Diagram, then export CSV or PDF.

Professional Notes

Why Bifurcation Diagrams Matter

Bifurcation diagrams summarize how long‑run behavior changes as a control parameter varies. They are common in nonlinear optics, population dynamics, electronic oscillators, and feedback control because a single graphic reveals stability loss, periodic orbits, chaotic bands, and periodic windows.

Logistic Map as a Benchmark

The logistic map is a canonical discrete‑time model with a single smooth nonlinearity. With the standard form, the nonzero fixed point is stable for 1 < r < 3, then undergoes period doubling at r = 3. The cascade accumulates near r ≈ 3.5699, and the spacing of bifurcations follows the Feigenbaum scaling that appears across many systems.

Tent Map and Piecewise Stretching

The tent map is piecewise linear, making it useful for understanding stretching and folding without heavy algebra. In many conventions, r is most informative in the range 0 to 2, where the map can be chaotic and quickly approaches an invariant distribution. This makes it a practical comparison case when teaching deterministic randomness and mixing.

Parameter Sweep Strategy

Choosing r-steps sets the horizontal resolution. For exploratory work, 600 to 1200 steps often shows the overall structure. For narrow windows, more steps are needed because periodic bands can be thin in r, sometimes narrower than 0.001. If you increase r-steps, consider reducing sample points to keep render time reasonable.

Burn-in and Transients

Early iterations depend strongly on the initial condition and may not represent the attractor. Burn-in discards these transients so that plotted points approximate steady behavior. Typical values are 300 to 1500 iterations, but regions near bifurcations can converge slowly. If the diagram looks smeared where it should be crisp, increase burn-in first.

Sampling and Vertical Density

Sample points control how many x values are plotted per r. Too few samples can miss higher‑period orbits and under‑represent chaotic bands. Too many samples can slow plotting and exports, so the calculator automatically thins points when the total becomes very large. A practical starting point is 80 to 200 samples per r.

Numerical Considerations

Chaotic trajectories are sensitive to tiny perturbations, including floating‑point rounding. That means two machines can generate slightly different point clouds even with identical settings, while still showing the same qualitative structure. Using an x0 not extremely close to 0 or 1 avoids trivial behavior and improves consistency in the logistic map.

Interpreting Common Patterns

Single lines indicate stable fixed points; two lines indicate a stable 2‑cycle. A dense vertical band indicates chaos, while isolated islands inside chaos indicate periodic windows. Sudden widening or splitting of bands can signal crises or band‑merging events. Comparing logistic and tent maps highlights how smooth versus piecewise dynamics shapes stability.