Consider a discrete time dynamical system

with an equilibrium value

Theorem: The equilibrium value is stable or attracting if

and is unstable or repelling if

If our work is in-conclusive.

The derivative of a curve at a point is, in some sense, the best linear approximation of the curve at that point.

• Worked Example 2

Consider the dynamical system

After making substitutions and we consider a curve

The equilibrium values which are the solutions of the equation are and Evidently,

Since if follows that the equilibrium value 2.75 is unstable (since ).

Likewise is unstable since If you were to construct a cobweb starting with a small positive initial value (say ), you would find that the solution would be repelled from