The framework of chaos is something which is fairly well defined in classical physics.
I am not exactly sure what you are aiming at here, but I fail to see any that chaos theory puts any emphasis on classical physical systems.
First of all, chaos is a mathematical phenomenon that can be seen in all sorts of dynamical systems (i.e., systems of differential equations) and thus chaos theory is somewhat ignorant of the physical background of a system.
Second, most chaotic systems are not even physical in the sense that they do not have something like energy conservation, e.g., they are models of fuelled chemical reactions, population dynamics, or the weather.
Do we have something like chaotic solutions of general relativity?
As general relativity approximates classical mechanics on everyday scales, you can start with something as simple as the double pendulum. There is no reason to assume that it turns non-chaotic if general relativity is taken into account. In fact, if this were the case, we would have to doubt general relativity as an accurate description of reality. After all, we can also experimentally confirm the chaoticity of the double pendulum.