bfd3e6f0-9dd4-4609-b548-785927997ba220251017055911246wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON BUSINESS AND ECONOMICS2224-28991109-952610.37394/23207http://wseas.org/wseas/cms.action?id=40161720251720252210.37394/23207.2025.22https://wseas.com/journals/bae/2025.phpK.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 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.101720251017202522872295180https://wseas.com/journals/bae/2025/d645107-051(2025).pdf10.37394/23207.2025.22.180https://wseas.com/journals/bae/2025/d645107-051(2025).pdfThom, R., Modèles Mathématique de la Morphogénèse. L’Enseignement Mathèmatiquet. VIII, 1962, pp. 24-33. Thom, 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. 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. 10.1016/s0960-0779(03)00132-2T. Puu and I. Suchko, A business cycle model with cubic nonlinearity, Chaos, Solitons & Fractals, 19, 2004, pp. 597-612. 10.1016/s0960-0779(02)00679-3I. Sushko, T. Puu, L. Gardini, The Hicksian floor-roof model for two regions linked by interregional trade, Chaos, Solitons & Fractals, 18, 2003, pp. 593-612. R.M.Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica, 19, 1951, pp. 1-17. H.-W. Lorenz, Goodwin’s nonlinear accelerator and chaotic motion, J.Economics, 2, 1987, pp. 315-318. H.-W. Lorenz, H.E. Nusse, Chaotic attractors, chaotic saddles and fractal basin boundaries: Goodwin’s nonlinear accelerator model reconsidered, Chaos, Solutions & Fractals, 13, 2002, pp. 957-965. T. Puu, Nonlinear Economic Dynamics, Lecture Notes in Economics and Mathematical Systems, 336, 1989, Springer-Verlag.