Aymptotically stable?

1. Apr 6, 2006

Tony11235

Suppose F and G are $$c^2$$ and $$F_x = F_y = G_x = G_y = 0$$ at the origin. Must the origin be an asymptotically stable equilibrium point?

One more

Give an explicit example of a DE with exactly two saddle points and no other equilibria. Anybody? Could I work backwards starting with the eigenvalues to form a system that has the two saddle points? This might be a dumb question.

2. Apr 7, 2006

HallsofIvy

Staff Emeritus
It would help a lot if you would actually state the problem clearly.
Without the requirement that dx/dt= F(x,y) and dy/dt= G(x,y), which you don't say, the problem makes no sense at all. Given that, what about F(x,y)= G(x,y)= 1. Is the origin even an equilibrium point?

3. Apr 7, 2006

Tony11235

This was just posed as a review question on a "things to know sheet". There weren't any specific details. Oh well. Too late. Test in 40 minutes.

4. Apr 7, 2006

HallsofIvy

Staff Emeritus
Presumably, then, you were expected to know what material was being reviewed or at least what course this is- things WE do not know!

5. Apr 7, 2006

Tony11235

Did you REALLY have to state the obvious?

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