On Relative Stability in 4-Dimensional Duck Solution

作     者：Kiyoyuki Tchizawa 

作者机构：Executive Dr. KanriKogaku Kenkyusho Ltd. (Institute of Administration Engineering Ltd.) Tokyo 101-0021 Japan 

出 版 物：《Journal of Mathematics and System Science》 (数学和系统科学（英文版）)

年 卷 期：2012年第2卷第9期

页      面：558-563页

学科分类：081704[工学-应用化学] 07[理学] 08[工学] 0817[工学-化学工程与技术] 070104[理学-应用数学] 0703[理学-化学] 070301[理学-无机化学] 0701[理学-数学] 

主　　题：相对稳定性 鸭解 不变流形 局部模型 矢量场 吸引力 奇异点 系统 

摘      要：This paper gives the existence of a relatively stable duck solution in a slow-fast system in R2+2 with an invariant manifold. The slow-fast system in R2+2 has a 2-dimensional slow vector field and a 2-dimensional fast vector field. The fast vector field restricts a feasible region of the slow vector field strictly. In the case of the slow-fast system in R2+1, that is, the fast vector field is 1-dimension, it is classified according to its sign, because it gives only negative(-), positive(+) or zero sign. Then it is attractive, repulsive or stationary. On the other hand, in R2+2, the fast vector field has combinatorial cases. It causes a complex state to analyze the system. First, we introduce a local model near the pseudo singular point on which we classify the fast vector field attractive(-,-), attractive-repulsive(-,+) or repulsive(+,+), simply as possible. We prove the existence of a 4-dimensional duck solution in the local model. Secondarily, we assume that the slow-fast system has an invariant manifold near the pseudo singular point. When the invariant manifold has a homoclinic point near the pseudo singular point, we show that the slow-fast sytem has a 4-dimensional duck solution having a relatively stable region.