📉 Logistic Map

It is the iteration x_n+1 = r·x_n·(1−x_n), where x is a value between 0 and 1 (often modelling a population fraction) and r is a growth-rate parameter between 0 and 4. Despite its simplicity, repeatedly applying this one formula produces everything from stable equilibria to periodic cycles to full chaos, depending only on the value of r.

For each r value along the horizontal axis, the simulation iterates the map many times, skips an initial run of transient iterations so the orbit settles down, and then plots the surviving x values as points stacked vertically. Where the map settles to one value you see a single point; where it settles into a cycle you see a small number of branches; where it is chaotic you see a dense smear, colour-coded by how often each pixel is visited.

As r increases past 3, the single stable equilibrium splits into a 2-cycle, then a 4-cycle, then an 8-cycle, and so on — a period-doubling cascade. The r values where these splits happen get closer together geometrically, and the whole cascade accumulates at r ≈ 3.56995, known as the Feigenbaum point. Beyond it the orbit no longer settles into any finite cycle and the dynamics become chaotic, though narrow periodic windows (like period-3 near r ≈ 3.83) still appear.

The cobweb view draws the parabola y = r·x·(1−x) together with the diagonal y = x, then traces the iteration step by step: a vertical line from x_n up to the parabola gives x_n+1, and a horizontal line to the diagonal carries that value back onto the x-axis for the next step. Watching this staircase-like path settle onto a point, bounce between a few points, or wander erratically shows visually whether the current r produces stability, periodicity or chaos.

The Lyapunov exponent λ measures how fast two nearby starting points diverge (or converge) under repeated iteration of the map. When λ is negative the orbit is stable or periodic, when λ is very close to zero the system sits at the edge of chaos (often right at a period-doubling point), and when λ is positive the dynamics are chaotic — tiny differences in x₀ grow exponentially, which is the hallmark of sensitive dependence on initial conditions.