$$\begin{split}
X_{t+1} = r X_t (1 - X_t) \\ \mbox{ with } 0<X_t<1 \mbox{ and }0<r<4
\end{split}$$ Solving the equation for $X$ as a function of $t$ we have $$X^{*} = 1-1/r$$

This solution is stable for $r \le 3$; otherwise it is oscillatory. For $r > 3.57$ approximately, the oscillations are infinitely complex; i.e. the dynamics are chaotic. Since the solution cannot be written explicitly, to see what it looks like we calculate successive values of $X$, while recognizing that these values become increasingly approximate, and soon become entirely arbitrary — that is, they become unpredictable even though the equation generating them is deterministic. This can be dismissed as simply due to the rounding errors that result from the finite precision of the computer, but it is actually a consequence of the interaction of the rounding errors with the fractal nature of the attractor. The rate at which the values evolve from precise to arbitrary is described by the Lyapunov exponent. So while the system may look well defined and timeless from an analytical point of view because the attractor determines the complete behaviour of the system, in fact the attractor is unknowable analytically, and can only be known 