There's a field of economics I've worked with called complexity economics that deals with this type of nonlinear behavior. My understanding of chaos theory is limited compared to other areas, but complexity economics focuses on nonlinearity, feedback loops, critical behavior, etc in economics. I know this relates to chaos theory, but as far as I understand isn't a direct application of it. It's hard to directly isolated the applications of chaos theory within this broader complexity economics framework (at least at the level of understanding of chaos I have).
The most important application I can think of are the impacts of small changes in initial conditions. Tiny changes can magnify into large scale macro changes ( the "straw that broke the camel's back" on a market crash that may not have happened otherwise, or a world changing invention that maybe just missed out on VC funding).
Another is multi state disequilibrium. Most economic models assume linearity and equilibrium, but complexity economics assumes a nonlinear, far from equilibrium system operating around attractors than can radically shift when critical points are approached. This applies to development/growth theories (in terms of pushing under developed nations to high growth states), or the boom bust cycle, etc. While this doesn't tell us much that hasn't been inferred elsewhere in economics, the tools of chaos/complexity allow us to study it more scientifically. Again, I'm not sure where the line switches from complexity to chaos systems, so apologies if I'm off.
Some of the more abstract mathematical elements (I'm thinking of bifurcations, fractals) I can't see having many economic applications, but someone else might in the future.
I've taken a class related to this, and am taking more, so I'll come back and add as more occurs to me. However sensitivity to small changes, criticality, nonlinearity, and multi-state disequilibrium are the big ones.