I have written a paper about the application of interval arithmetic to the problem of fractal rendering (if one can call it a "problem"). The paper is available here:
Drawing Fractals With Interval Arithmetic - Part 1
The paper includes a fractal-explorer tool for reproducing the results, a sample of which can be seen here:
The graphic on the left is a Mandelbrot drawn with the traditional single-point approach, and the one on the right is the result of the interval approach.
Like fractals, many (if not all) chaotic systems are iterative, and this paper has implications for simulating chaotic systems regardless of whether one uses interval or single-point methods.
In Part 2 I will try and quantify the results I am seeing, and investigate more sophisticated implementations of the maths behind the Mandelbrot equation.