The logistic map
The logistic map x → rx(1 − x): fixed points, when they attract and when they repel, and angle doubling — the simplest example of chaos.
How to solve it
The logistic map $x \mapsto rx(1 - x)$ is a simple model of a population, and chaos is born in it. A fixed point is a value that does not change. Whether it is stable is decided by the slope of the graph there: if its absolute value is below one, nearby values are drawn in.
Step by step
- Fixed points: solve $f(x) = x$. For $f(x) = rx(1 - x)$ they are $x = 0$ and $x^* = 1 - \frac{1}{r}$.
- The slope: $f'(x) = r(1 - 2x)$; at the non-zero fixed point $f'(x^*) = 2 - r$.
- Stability: $|f'(x^*)| < 1$ means the point attracts, $|f'(x^*)| > 1$ that it repels. For the logistic map that is $1 < r < 3$.
- Angle doubling $\theta \mapsto 2\theta \pmod 1$: double and drop the integer part until the number repeats; the number of steps is the period.
Common mistakes
- Forgetting that $x = 0$ is a fixed point too, or answering zero when the non-zero one is asked.
- Comparing the slope itself with one instead of its absolute value: at $r = 4$ the slope is $-2$ and the point is unstable.
- Not dropping the integer part when doubling an angle.
- Taking the first repeat of any value as the period, instead of the return of the starting one.