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,
orbits starting at nearby points can diverge from each other exponentially.
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.
the truth is that
rounding errors affect numerical simulations of dynamical systems in
very complex ways .
Well-conditioned dynamical systems may display chaotic numerical behavior [2,3].
numerical methods can suppress chaos in some chaotic dynamical systems .
(Text extracted from .)
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.
 Corless, What good are numerical simulations of chaotic dynamical systems? Comput. Math. Appl. 28 (1994) 107–121. MR1300684
 Adams et al., Computational chaos may be due to a single local error. J. Comput. Phys. 104 (1993) 241–250. MR1198231
 Corless et al., Numerical methods can suppress chaos, Physics Letters A 157 (1991) 127-36. DOI 10.1016/0375-9601(91)90404-V
 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