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
['https://math.stackexchange.com/questions/333882', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/62659/']
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](http://www.ams.org/mathscinet-getitem?mr=1300684) 

[2] Adams et al., Computational chaos may be due to a single local error. *J. Comput. Phys.* **104** (1993) 241–250. [MR1198231](http://www.ams.org/mathscinet-getitem?mr=1198231)

[3] Corless et al., Numerical methods can suppress chaos, *Physics Letters A* **157** (1991) 127-36. [DOI 10.1016/0375-9601(91)90404-V](http://www.sciencedirect.com/science/article/pii/037596019190404V)

[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](http://www.sciencedirect.com/science/article/pii/S0097849306001543)
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.