3.700
0.200
Chaotic regime: sensitive dependence on initial conditions, although narrow periodic windows exist.
xₙ₊₁ = r xₙ(1 − xₙ)
Interpretation: x is a normalized population or activity level from 0 to 1.
The parameter r controls growth plus density-dependent limitation. Small parameter changes can radically change long-term behavior.
Cobweb diagram
The parabola is f(x)=rx(1−x); the diagonal is y=x. The staircase shows repeated iteration.
Time series
Watch whether the sequence settles to one value, alternates, cycles through multiple values, or behaves irregularly.
Bifurcation diagram
Each vertical slice shows the long-term values of x for a given r. This reveals the classic route: fixed point → period doubling → chaos, with periodic windows embedded inside chaos.
Key regimes
| Approximate r range | Typical behavior |
|---|---|
| 0 < r < 1 | Population decays toward 0. |
| 1 < r < 3 | Stable non-zero fixed point. |
| 3 < r < 3.449... | Stable period-2 oscillation. |
| 3.449... < r < 3.544... | Period-4 oscillation, then further doublings. |
| ≈ 3.57 and above | Mostly chaotic, with islands of periodic order. |