Digital computers use finite precision arithmetic, which means that they can only represent numbers with a finite number of digits. This can lead to errors when modeling chaotic systems, which are often characterized by very small differences in initial conditions leading to large differences in long-term behavior.

To illustrate this, consider the following simple chaotic system:

$$\begin{equation}

x_{n+1} = 4x_n(1-x_n)

\end{equation}$$

where $x_n$ is the state of the system at time $n$. If we simulate this system using a computer with finite precision arithmetic, we will inevitably introduce errors into the calculation of $x_n$. These errors will grow over time, eventually leading to large differences between the simulated and actual behavior of the system.

The accuracy of a digital computer simulation of a chaotic system can be improved by using higher precision arithmetic, but this comes at the cost of increased computational time and memory usage. In some cases, it may be necessary to use special techniques, such as adaptive step size control, to ensure that the errors remain small enough to not significantly affect the results of the simulation.