I think the key idea behind all of nonlinear systems is that of feedback loops, I think nonlinear systems are really governed by the interaction between their parts and the feedback loops over time. Closed-form, static equation-based models are really not the appropriate tool. Because of feedback, you get emergence and you have to actually compute the whole thing to see what happens, you can’t know it beforehand. The optimal way to model emergence and a nonlinear system is then to simulate it. So I think the answer to your question is that the link between the models and the “real world” is simulation. You write the rules, compute it, and you get a visualization of what it is like.