On a Circuit Model proving the existence of 4-dimensional Canards with co-dimension 2 [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Kiyoyuki Tchizawa
Tokio City Univerisity


This talk gives the existence of a relatively stable duck solution in a slow-fast system in $R_{2+2},$ with an invariant manifold. The slow-fast system in $R_{2+2}$ 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 $R_{2+1}$ , 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 $R_{2+2},$ 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.