I don't quite get how chaos theory changed the world. All of the other equations I get (although whether Shannon's equation changed the world or merely described the change is probably arguable.)
Understanding deterministic chaos has been the cornerstone of a wide range of fields, including physics, medicine, economics, and business. There are a wealth of everyday applications. Control systems in the real world have to deal with the underlying structure of the dynamics that govern how such systems naturally behave in the feedback loop.
There are a few different resources you can find that list practical applications of chaos theory. Here are a few that I found that I think you will find interesting:
It has very important epistemological implications: it means a phenomenon may follow a deterministic law which, due to its mathematical properties, would nevertheless make long-term predictions impossible. So even though you would have a mathematical model for the phenomenon, you wouldn't be able to make real-world long-term predictions. The problem lies in the sensitive dependence on initial conditions.
How did chaos theory change the world?