cb117d7f-aaa5-44d3-a952-ba87c07e4a8a20250715063833832wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051120202512020252410.37394/23206.2025.24https://wseas.com/journals/mathematics/2025.phpCatastrophe with Brownian Motion on 4-dimensional CanardK.TchizawaInstitute of Administration Engineering, Ltd. 2-2-2 Sotokanda, Chiyoda-ku,Tokyo 101-0021 JAPANS.KanagawaDepartment of Mathematics Tokyo City University 1-28-1 Tamazutsumi, Setagaya-ku,Tokyo 158-8557 JAPANOn 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.7152025715202548249247https://wseas.com/journals/mathematics/2025/a945106-024(2025).pdf10.37394/23206.2025.24.47https://wseas.com/journals/mathematics/2025/a945106-024(2025).pdfThom, R., La Stabilité Topologique de Applications Polynomiales. L’Enseignement Mathèmatiquet. VIII, 1962, pp. 1-2. Thom, R., Stabilité Structurelle et Morphogénèse. Benjamin, New York, 1972. Thom, R., Modèles Mathématique de la Morphogénèse., Union Gènèrale d’Editions, 1974. Poston, T. and Stewart, I., Catastrophe Theory and its Applications, Pitman Publishing Ltd., 1978. 10.4236/apm.2022.1211046Kanagawa, S. and Tchizawa, K., Structural Stability in 4-Dimensional Canards, Advances in Pure Mathematics, Vol. 12, 2022, pp. 600-613. 10.1088/1361-6544/ad6bdeHildeberto, J., Christian, K., and Maximilian, S., The hyperbolic umbilic singularity in fast-slow systems, Nonlinearity, Vol. 37, No.9, Paper No. 095036, 2024. Mather, J., Stability of C∞-mappings I. The division theorem, Ann. Math., Vol. 87, 1968, pp. 89-104. 10.2307/1970668Mather, J., Stability of C∞-mappings II. Infinitesimal stability implies stability, Ann. Math., Vol. 89, 1968, pp. 254-291. 10.1007/bf02698926Mather, J., Stability of C∞-mappings III. Finitely determined map germs, Publ. Math. IHES., Vol. 35, 1968, pp. 127-156. 10.1007/bf02684889Mather, J., Stability of C∞-mappings IV. Classification of stable germs by R-algebras, Publ. Math. IHES, Vol. 37, 1969, pp. 223-248. 10.1016/j.nonrwa.2020.103286Meza-Sarmiento, I., Oliveira, R. and Da Silva, P., Quadratic slow-fast systems on the plane, Nonlinear Analysis in Real World & Applications, Vol. 60, 2019, Paper No. 103286. 10.1007/s10883-016-9335-6Schurov, I. and Solodovnikov, N., Duck factory on the two-torus: Multiple Canard Cycles without Geometric Constraints, Dynamical Control Systems, Vol. 23, 2017, pp. 481-498. 10.1007/s40863-024-00441-8Perez, O. and Da Silva, P., Polynomial slow-fast systems on the Poincaré-Lyapunov sphere, São Paulo J. Math. Sci., Vol. 18, 2024, pp. 1527-1552.