%0 Journal Article
%T Hyper Catastrophe on 4-Dimensional Canards
%A Kiyoyuki Tchizawa
%A Shuya Kanagawa
%J Advances in Pure Mathematics
%P 196-203
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.144011
%X When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion ¡°symmetry¡± plays an important role. It is of one parameter on slow vector field. Then, it should be determined to introduce parameters to all slow/fast vectors. It is, however, there might be no way to explore for another potential in this system, because the geometrical structure is quite different from the system with one parameter. Even in this system, the ¡°symmetry¡± is also useful to obtain the potentials classified by R. Thom. In this paper, via the coordinates changing, the possible way to explore for the potential will be shown. As it is analyzed on ¡°hyper finite time line¡±, or done by using ¡°non-standard analysis¡±, it is called ¡°Hyper Catastrophe¡±. In the slow-fast system which includes a very small parameter <img src=\"https://html.scirp.org//file/5302411-rId15.svg?20240407033445\" />, it is difficult to do precise analysis. Thus, it is useful to get the orbits as a singular limit. When trying to do simulations, it is also faced with difficulty due to singularity. Using very small time intervals corresponding small <img src=\"https://html.scirp.org//file/5302411-rId15.svg?20240407033445\" />, we shall overcome the difficulty, because the difference equation on the small time interval adopts the standard differential equation. These small intervals are defined on hyper finite number <i>N</i>, which is nonstandard. As <img src=\"https://html.scirp.org//file/5302411-rId15.svg?20240407033445\" /> and the intervals are linked to use 1/<i>N</i>, the simulation should be done exactly.
%K Canards Flying
%K 4-Dimensional Slow-Fast System
%K Hyper Catastrophe
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=132326