The only model that can model a complex non-linear system is the system itself. Climate system is complex because it has feedbacks. Heat transfer is governed by non-linear equations.
Complex non-linear systems are very sensitive to initial and boundary conditions. It would not use the word chaos because it is often connected to randomness. Lottery balls are a typical example of these systems. System is governed by deterministic laws of physics but small changes in the conditions result in unpredictable outcomes that in this case can be statistically analysed.
I spent a lot of time to think about Nick Stoke’s claim that GCMs are physical models that do not use weather station data. Behind his link to a simple climate model I found that initial and boundary values for each grid cell are used (look at chapter 7). I would call that deception because calculating grid cell values from weather station data is of course needed.
Googling ahead, I found a number of references that claim the climate system is not sensitive to iniatial situation. Just above in this thread: ”Run different meterological models out far enough in time, and they ALL eventually diverge”. I am not sure of that. Starting at a ice age and at hot period in history will result in very different next 100 years. Sensitivity to initial data is definitely there but not necessarily to the athmospheric data but to the data of ocean heat and ice.
Missing understanding of the clouds and oceans is a showstopper in climate modelling. Without accurate and precise data it is not possible to get that understanding. Think about where HITRAN has been defined.
By definition models are simplifications of the system. You have to leave out unimportant factors and accuracy & precision have to be compromized because of limits of the computing power. Numerical calculation has its limits. Current climate models have overly large grid cells and weak parametrization to correct that. Overcoming that in a foreseable future is not probable.