TITLE: Are continuous chaotic systems necessarily uncomputable?
QUESTION [9 upvotes]: I have seen the claim in a recent unpublished paper that chaotic dynamics are necessarily uncomputable. This follows, they argue, from the sensitivity to initial conditions shown in chaotic systems. This glib identity between chaos and computability struck me as simplistic, but I have not found a good discussion of the topic that I trust.
I found one discussion by  Svozil (1989) 'Are chaotic systems dynamically random?' which claims there exist four types of chaos, depending on whether the initial conditions are computable, and whether the evolution of the system is computable. This generates four possibilities: for instance Chaos of Type I is "generated by a computable evolution of a system with
uncomputable...initial values" (p 4). The fourth type is the chaotic dynamics produced when both initial conditions and evolution of the system are computable. This would suggest that chaos does not imply uncomputability (of either type).
Is this division of possibilities generally accepted? More to the point, is the relationship between chaos and computability worked out sufficiently so there is now an obvious answer to my question?
Reference
Karl Svozil. Are chaotic systems dynamically random?, Phys. Lett. A 140, 5-9 (1989)
http://tph.tuwien.ac.at/~svozil/publ/1989-dyn.pdf
Edit 1
I should have been more clear that my question is about chaos in continuous systems, not discrete systems.
Also, after asking this question I found the following potentially relevant papers:

Corless (1994) What good are numerical simulations of chaotic
dynamical systems? Computers Math. Applic. 28: 107-121.
Yao (2010) Computed chaos or numerical errors. Nonlinear Analysis
2010: 109-126.
Lia (2014) A comment on the arguments about the reliability and
convergence of chaotic simulations. http://arxiv.org/abs/1401.0256v1.

Edit 2
Via discussions below, there is one subquestion that has emerged that seems to identify the crux of the issue with more precision than my original question:

Assume (a) that any numerical simulation of the solution of continuous
chaotic dynamical system X will necessarily strongly diverge from the
actual solution (at some point in time in the simulation).
Is it true, in any of the well-defined senses of 'computable' from the
theory of computation, that the behavior of system X is uncomputable?
If no, then the claim that chaotic systems are uncomputable, in the
sense usually intended, seems false as a mathematical claim.
If yes,
then two subquestions. 1. For what senses of computable is the
system uncomputable? 2. Is assumption (a) true for any interesting scientific model
(e.g., a specific model from neuroscience that people actually use)?

Edit 3
Still waiting for an answer to this question. I did get a response from a philosopher who didn't answer it but suggested a book. Here is his message to me:

You may want to have a look at the book "Complexity and Real
Computation" by Blum, Cucker, Shub & Smale. I still have to real most
of it, but I think you will find their analysis of computability
helpful. Their don't specifically address chaotic dynamics, but they
thoroughly analyze the relation between condition numbers (chaotic
problems are tough precisely because they're ill-conditioned),
complexity, and computability (in Turing's sense and for arbitrary
rings and fields).

I haven't looked at the book yet.

REPLY [4 votes]: Choaticity in computable systems are a well known topic in theoretical computer science. For example, you can read numerous articles on cellular automata and their computational behaviour. A chaotic behaviour will be defined using some usual topology and it can be proved that some cellular automata are chaotic (it means there are some butterfly effects). But they are entirely computable, as well as initial conditions.
From a purely computability point of view, chaoticity can appear as soon as you can't predict some behaviour. Many problems are computable but some behaviour (like the halting problem) can't be predicted.
An example of that is the n-body problem in physics. Computations can be made to define the next state of the system, but you can't answer some question like "Will this planet  crash into this other one ?"
So, to my point of view, chaotic systems can be computable, but you can ask uncomputable questions about them.