Abstract: On the 4-dimensional canards having Brownian motion, we provide that there exists the rigidity for the motion from the out side. Furthermore, in the FitzHugh-Nagumo without Brownian motion, being as a concrete model, it is confirmed that there exists ”catastrophe” as a potential ”hyperbolic umbilic”. In the two-region economic model, it does 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 though. The reason is complex nonlinearity. Even in this system without Brownian motion, the model has a potential such as ”hyperbolic umbilic”. Corresponding simulations are done by Nonstandard Analysis using the hyper finite timeline and small intervals. Then, the difference system discretizing by using the nonstandard number 1/N has different potentials on each interval created by Brownian motion. In this paper, how to construct the catastrophe caused by Brownian motion will be described.

Keywords: Canards flying, 4-dimensional slow-fast system, Hyper catastrophe

Pages: 482-492