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＋： has a 2-dimensional slow vector field and a 2-dimensional f...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＋： 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 l-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.展开更多
文摘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＋： 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 l-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.
