I am trying to use this result but want to have some justification of it. I would like to explain in terms of limits sets of the saddle's stable and unstable manifolds.
Asked
Active
Viewed 119 times
0
-
Are you talking about planar systems? – Evgeny Nov 19 '15 at 18:33
-
@Evgeny yes $R^2$ – grayQuant Nov 19 '15 at 18:37
-
The fastest way is to use index theorem. Index along periodic orbit equals 1, but index of hyperbolic saddle is -1. You need a +2 of index to compensate this, which means that at least 2 sources or sinks are needed for this -> extra equilibria. – Evgeny Nov 19 '15 at 18:43
-
@Evgeny thanks, can you think of some other explanation just discussing the manifolds? I will look up index theorem – grayQuant Nov 19 '15 at 18:45
-
Well, you have two of them, stable and unstable. And if one of them is asymptotic to limit cycle other should be asymptotic to something else. Hence the need for "something else". – Evgeny Nov 19 '15 at 19:16