Chaos theory and determinism

Well, yes. In a purely mathematical world where you can specify initial conditions exactly, chaotic systems are fully deterministic. It's not like a quantum system with wavefunction collapse, whose evolution can never be specified exactly by the initial conditions.

But in practice, we can never specify (or know) the initial conditions exactly. So there will always be some uncertainty in the initial conditions, and it makes sense to characterize the behavior of a system in terms of its response to this uncertainty. Basically, a chaotic system is one in which any uncertainty in the state at time $t=0$ leads to exponentially larger uncertainties in the state as time goes on, and a non-chaotic system is one in which any initial uncertainty in the state decays away or at least stays steady with time.

In the former (chaotic) case, given that we can't know the initial conditions to infinite precision, there will always be some time after which predictions of the behavior of the system become essentially meaningless - the uncertainty becomes so large that it fills up most of the state space. This is effectively similar to the behavior of a truly non-deterministic (e.g. quantum) system, in that our ability to make predictions about it is limited, so some people call chaotic systems non-deterministic.

Classical mechanics is perfectly integrable for two bodies as a closed or isolated system. However, early on it was found that problems existed, where Newton found he could not find a solution for the motion of the planets in a complete form. He made his famous statement that God had to readjust the solar system now and them. Poincare solved the Sweden prize for a solution to the stability of the solar system by demonstrating no such solution existed. This is what opened the door for chaos theory, where Poincare developed methods of separatrices and perturbations on them. For general systems it turns out that Newtonian mechanics is not integrable,

Classical mechanics for systems with three or more bodies can\rq t be solved in closed form. For any $N$ body problem there are 3 equations for the center of mass, $3$ for the momentum, $3$ for the angular momentum and one for the energy. These are $10$ constraints on the problem. An N-body problem has $6N$ degrees of freedom. For $N~=~2$ this means the solution is given by a first integral with degree $2.$ For a three body problem this first integral has degree $8$. This runs into the problem that Galois illustrated which is that any root system with degree $5$ or greater generally have no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for $8$ solutions there is some eight order polynomial $p_8(x)~=~\prod_{n=1}^8(x~-~\lambda_n)$, with $8$ distinct roots$\lambda_n$ that are constant along the $8$ solutions. Since $p_8(x)~=~p_5(x)p_3(x)$, a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding its roots, or a set of solutions that are algebraic. This means that any system of degree higher than four are not in general algebraic. At the root of the $N$-body problem Galois theory tells us there is no algebraic solution for $N~\ge~3$.

A problem in classical mechanics is the vanishing denominator problem for three bodies. This has a Hamiltonian $H( J,~\theta)~=~H_0(J,~\theta)~+~\epsilon H_1(J,~\theta)$ for $J~=~(J_1,~J_2)$ and $\theta~=~(\theta_1,~\theta_2)$. Here a generating function written according to the variable $J^\prime$
$$ S(J^\prime,~\theta)~=~\theta\cdot J^\prime~+~i\epsilon\sum_{n_1,n_2}{{H_{1,n_1,n_2}}\over{n_1\omega_1(J^\prime)~+~n_2\omega_2(J^\prime)}}e^{ n_1\omega_1(J^\prime)~+~n_2\omega_2(J^\prime)} $$ will be divergent for the resonant condition $n_1\omega_1(J^\prime)~+~n_2\omega_2(J^\prime)~=~0$. This resonance condition has lead many to presume that the solar system can not be stable with resonance conditions. Yet the solar system is replete with near resonance conditions.

The number of resonance conditions that exist on the real line are dense. Within any $\epsilon$ neighborhood there will exist a countably infinite number of possible resonance conditions that correspond to rational numbers. As the orbit of a planet drifts it will pass through these resonance conditions and be chaotically perturbed. It is to be expected that for simple rational numbers, such as $1/12$ for Earth and Jupiter, rather than $1003/12000$ strong resonances occur. For more complex rational numbers it might be expected that the instability will be weaker. In other words if the ratio of frequencies are \lq\lq sufficiently irrational\rq\rq$~$so that $$ \Big|{{\omega_1}\over{\omega_2}}~-~{m\over s}\Big|~>~{{k(\epsilon)}\over{s^{2.5}}},~\lim_{\epsilon~\rightarrow~0}k(\epsilon)~\rightarrow~0 $$ the orbit is more stable. So an orbit that is removed from a “strong resonance”” condition near a simple rational number will be more stable than an orbit that is near an orbit with a simple rational ratio of frequencies.

This is the basis for Greenberg’s Hamiltonian approach to chaos theory. This is called deterministic for the differential equations are time reverse invariant so the motion of a particle is absolutely determined. However. if you have a slight variation in the initial conditions of that particle is may in general end up arbitrarily far away from its starting point. The tiny variation $\delta z~=~(\delta q,~\delta p)$ becomes amplified by an exponential map $\delta z~\rightarrow~exp(\lambda t)\delta z$, for $\lambda$ the Lyapunov exponent. Any error in the specification of the initial conditions of a body results in the amplification of this error. From an algorithmic perspective a truncation results in numerical overflow errors which grow. So the dynamics of a particle can’t be integrated by computer to arbitrary accuracy into the future, even though nature actually does determine its dynamics.

W. Zurek took this a bit further and considered how quantum fluctuations, where $\delta z$ is set by the Heisenberg uncertainty principle.

I think the following from the wikipedia entry clears up well the terminology:

Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions; an effect which is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general.1 This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.[2] In other words, the deterministic nature of these systems does not make them predictable.[3] This behavior is known as deterministic chaos, or simply chaos.

Bold mine.

Note that chaos results in completely deterministic systems when small errors in initial conditions yield highly divergent solutions. It is the "exactly" in your question that is unattainable, that will be a tiny bit off in chaotic situations,(highly nonlinear response to input parameters) even in computer solutions, because one cannot be more accurate than the computer bits.

Note also that this does not mean that there are no mathematical methods to study the behavior of such systems. There are, and can be predictive in bulk. I would give as an example the study of Tsonis et al who have studied climate with a neural net chaotic model using as inputs bulk behavior of atmospheric and ocean currents.