Logistic Map

The logistic map is one of the simplest yet most fascinating examples of how complex, chaotic behaviour can emerge from a simple nonlinear equation:

$x_{n+1} = r \cdot x_n \cdot (1 - x_n)$

Here $x_n \in [0,1]$ represents a normalised population at generation $n$, and $r$ is the growth rate parameter.

Behaviour by Region

Fixed Point

For $r > 1$, the non-trivial fixed point satisfies $x^* = r x^* (1 - x^*)$, giving:

$x^* = 1 - \frac{1}{r}$

Lyapunov Exponent

The Lyapunov exponent $\lambda$ measures the average rate of divergence of nearby trajectories:

$\lambda = \frac{1}{N} \sum_{n=0}^{N-1} \ln \left| r(1 - 2x_n) \right|$

• $\lambda < 0$: stable (convergent) dynamics
• $\lambda > 0$: chaos (exponential divergence)

For the fully chaotic case $r = 4$, the Lyapunov exponent equals $\ln 2 \approx 0.693$.

Your Task

Implement three functions:

• logistic_iterate(r, x0, n) — iterate the map $n$ times from $x_0$, returning a list of $n$ values
• logistic_fixed_point(r) — return $x^* = 1 - 1/r$ for $r > 1$, else $0$
• lyapunov_exponent(r, x0=0.5, n=1000) — compute $\lambda$, skipping the first 100 transient steps