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

In summary, there is a rule of thumb used by researchers to determine the stability of fixed points in nonlinear systems. This rule states that if the product of the gradients of the null clines at the point of intersection is less than -1, the fixed point is considered stable. If the product is greater than -1, the fixed point is either neutrally stable or unstable. There is no formal proof for this rule, but one way to visualize it is by looking at the vector field of an attractor and a repeller and analyzing the angles of the vectors.
  • #1
Appaloosa
2
0
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..
 
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  • #2
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.
 

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

What is the significance of the product of gradients at the intersection of null clines in a 2D system?

The product of gradients at the intersection of null clines in a 2D system is a measure of the stability of the system. It determines whether the system will converge to a steady state or exhibit oscillatory behavior.

How is the product of gradients calculated in a 2D system?

The product of gradients is calculated by taking the partial derivatives of the equations that describe the null clines at the point of intersection. These partial derivatives are then multiplied together to obtain the product of gradients.

What do positive and negative values of the product of gradients indicate in a 2D system?

A positive product of gradients indicates that the system is stable and will converge to a steady state. On the other hand, a negative product of gradients indicates that the system is unstable and will exhibit oscillatory behavior.

Can the product of gradients be used to predict the behavior of a 2D system?

Yes, the product of gradients can be used to predict the behavior of a 2D system. It provides information about the stability of the system and can help determine whether the system will reach a steady state or exhibit oscillatory behavior.

What is the relationship between the product of gradients and the eigenvalues of the Jacobian matrix in a 2D system?

The product of gradients at the intersection of null clines is directly related to the eigenvalues of the Jacobian matrix in a 2D system. The sign of the product of gradients will determine the sign of the eigenvalues, which in turn determines the stability of the system.

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