Chaos theory — the logistic map

What just happened?

You ran the same formula for two nearly identical starts: $x_0 = 0.500$ and $x_0 = 0.501$. For small $r$ both runs coincide. But in the chaotic regime ($r \gtrsim 3.57$) they diverge completely after a few steps — even though the difference in the start was microscopic. This is the famous butterfly effect: the flap of a butterfly’s wings can change the weather thousands of kilometres away.

The logistic map:

It is a simple population-growth model: $x_n \in [0,1]$ is the population density at step $n$, and the parameter $r$ (the reproduction rate) determines the behaviour of the system.

From stability through period doubling to chaos:

As we increase $r$, we observe a transition: for $r < 3$ the system is STABLE — it converges to a single value. Above that it begins to OSCILLATE, doubling its period (2, 4, 8, 16… values) — this cascade of bifurcations was studied in 1978 by Mitchell Feigenbaum. Finally, at $r \approx 3.57$ it falls into CHAOS: the values jump irregularly, with no repeating pattern.

Chaos is not randomness:

The system is fully DETERMINISTIC — no randomness, the same $x_0$ always gives the same trajectory. And yet it is practically UNPREDICTABLE in the long run, because an arbitrarily small uncertainty in $x_0$ grows exponentially. Just like the weather, a double pendulum or the three-body motion. This was discovered (by accident, while studying a weather model) by the meteorologist Edward Lorenz in 1963.

Why this matters for your exams:

The logistic map is the prime example that a simple recursive sequence can behave in a stable, periodic or chaotic way — depending on a single parameter. The idea of “sensitivity to initial conditions” shows that simple rules do not guarantee predictability. Chaos theory revolutionised our understanding of nature.

Remember: chaos is deterministic, but unpredictable — $x_{n+1} = r\,x_n(1-x_n)$ shows this in a single formula.