Alfred Hubler: Accurate Low-Dimensional Discrete Models for Continuous Systems

Natural Sciences lecture series

Alfred Hubler discusses the relationship between CA and continuous systems

Cellular automaton are low-dimensional dynamical systems which are discrete and local in time and space and have discrete amplitude. Which high-dimensional continuous systems make good models as CA?

In order to find a low-dimensional model, the first step is to separate the fast and slow variables in the system; Since the fast variables typically decay rapidly, they are not of interest. This is usually accomplished by integration over one period of the fast motion, on both sides of the differential equation. Next, a flow vector field is approximated from the previous integro-difference equation. This gives a difference equation, which is then approximated by a differential equation, which are easy to integrate analytically.

Low-dimensional models are good if the separation of time scales is very large, e.g. in rigid-body motion but not for "soft-body" motion. The equations of motion can be improved through discretization, using Euler's method. Continuous-time motion equations are good if the time scale is very large.

Time-discrete models are accurate for both rigid and soft systems, while continuous systems are only accurate for rigid systems. Models in the form of difference equations are often qualitatively better than time continuous models with the same number of variables. Cellular automaton can be shown to be more accurate than PDEs in some situations.