Mathematical Limitations of Computer Experiments

Question

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

Answer 1

Numerical simulation of dynamic systems is indeed hard.

One difficulty is that it is implemented using floating-point arithmetic, which is subject to rounding errors. For chaotic dynamical systems, the ones that have strange attractors, rounding errors are potentially serious, because orbits starting at nearby points can diverge from each other exponentially. Sometimes, this strong sensitivity to initial conditions does not affect the overall picture, because numerically computed orbits are shadowed by exact orbits that capture the typical behavior of the system. However, the truth is that rounding errors affect numerical simulations of dynamical systems in very complex ways [1]. Well-conditioned dynamical systems may display chaotic numerical behavior [2,3]. Conversely, numerical methods can suppress chaos in some chaotic dynamical systems [3]. (Text extracted from [4].)

Nevertheless, there are computational methods for studying dynamical systems that work reliably even in the presence of rounding errors. See the work of Warwick Tucker, Zbigniew Galias, Pawel Pilarczyk, and others.

[1] Corless, What good are numerical simulations of chaotic dynamical systems? Comput. Math. Appl. 28 (1994) 107–121. MR1300684

[2] Adams et al., Computational chaos may be due to a single local error. J. Comput. Phys. 104 (1993) 241–250. MR1198231

[3] Corless et al., Numerical methods can suppress chaos, Physics Letters A 157 (1991) 127-36. DOI 10.1016/0375-9601(91)90404-V

[4] Paiva et al., Robust visualization of strange attractors using affine arithmetic, Computers & Graphics 30 (2006), no. 6, 1020-1026. DOI 10.1016/j.cag.2006.08.016

Answer 2

It's a valid concern, and I have a historical anecdote to match it. When Mandelbrot published the first picture of the Mandelbrot set, the image was of poor quality due to the state of informatics back then. Seeing what he thought where "speck of dusts" on the picture, his editor erased them. Since then we obtained much better pictures of course, and we know that these specks of dust are "islands" that are linked to the main body of the Mandelbrot set by "hairs" of empty interior (which were not very visible in Mandelbrot's picture).

The point is, computers are a tool to guide your intuition but you should not rely solely on them. Think of imperfect computer pictures/simulation as drawing a surface instead of a n-manifold : it helps understanding what is going on, but it works together with a more rigorous understanding of the objects you are manipulating. Otherwise you are indeed bound to make mistakes.

Answer 3

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.