DE with Two Saddle Points: Finding an Explicit Example

[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.

 

[HallsofIvy]

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?

 

[Tony11235]

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?

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.

 

[HallsofIvy]

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

 

[Tony11235]

Did you REALLY have to state the obvious?