Catastrophe in the Two-regional Economics

On 4-dimensional canards having Brownian motion, we prove that there exists the rigidity for the motion from out side. Furthermore, it is confirmed that there exists ”catastrophe” as a potential ”hyperbolic umbilic” by R. Thom. Then, we have a question: in the two-region economic model, does it also exist ”catastrophe”? Bifurcation for the pseudo singular point, which depends on some parameters has already been analyzed enough to realize it. In the two-regional economic model, it is, however, not yet analyzed still now though. The reason is complex nonlinearity, but the answer is yes. Even in this system, the model has a potential such as the hyperbolic umbilic. Why and how to construct corresponding potentials in order to catch up an orbit jumping by the bifurcation. In this paper, they will be described.