structural:reformatting
Maps

Let f: R → R be a continuously differentiable function with a fixed point a, f (a) = a. Consider the dynamical system obtained by iterating the function f:

The fixed point a is stable if the absolute value of the derivative of f at a is strictly less than 1, and unstable if it is strictly greater than 1. This is because near the point a, the function f has a linear approximation with slope f' (a):

Thus

which means that the derivative measures the rate at which the successive iterates approach the fixed point a or diverge from it. If the derivative at a is exactly 1 or −1, then more information is needed in order to decide stability.

There is an analogous criterion for a continuously differentiable map f: R → R with a fixed point a, expressed in terms of its Jacobian matrix at a, J (f). If all eigenvalues of J are real or complex numbers with absolute value strictly less than 1 then a is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then a is unstable. Just as for n =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.