TITLE: Mathematical Limitations of Computer Experiments
QUESTION [6 upvotes]: One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never repeat themselves. But computationally, it seems that such an orbit can never be realized (since given the finite number of values that a computer can work with, periodicity seems inevitable). Yet, people study chaos with computers. They use mathematics of course, but considering the role that computers play in helping us gain intuition of complex systems, I wonder if there are aspects of a system we might totally miss by using computers in such a fashion.
Thanks,
Jack

REPLY [2 votes]: The set of nonlinear systems that can be understood analytically is a measure zero set. Numerical experiments, and numerical algorithms, are a necessary tool in the study of these systems. But I agree that there is a need for more robust numerical tools for studying complex systems.
Anyways, regarding chaotic sets, the "set-oriented" Perron-Frobenius operator-theoretic approach to numerically studying the densities and probability distributions on these systems rather than individual trajectories has been quite successful. See work by G.Froyland, M. Dellnitz, I. Mezic etc. On the other hand, topological quantities such as topological entropy are robust to numerical errors and don't rely on exact computation of trajectories, e.g. see work by P. Boyland etc. So there are other ways of understanding these systems than just numerically tracking individual trajectories.