Product of gradients at intersection of null clines in 2D system

[Appaloosa]

Hi all,

it seems that there is a rule of thumb used by some researchers looking at nonlinear systems whereby they determine the stability of fixed points based on the product of the gradients of the null clines at the point where they intersect. in particular if the product of the gradients is < -1 the fixed point is assumed to be stable and if it is > -1 the fixed point is either neutrally stable or unstable. i can't find the proof of this result anywhere, does anyone know of a reference which discuss this result or know if this is a named theorem?

many thanks..

 

[fresh_42]

You cannot 'prove' a rule of thumb. An idea is to draw the vector field of an attractor and a repeller. Then have a look at the angles and whether they point out- or inwards, i.e. whether they are acute or obtuse.