Fixed Point (mathematics) - Attractive Fixed Points

Attractive Fixed Points

An attractive fixed point of a function f is a fixed point x₀ of f such that for any value of x in the domain that is close enough to x₀, the iterated function sequence

converges to x₀. An expression of prerequisites and proof of the existence of such solution is given by Banach fixed point theorem.

The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, which is attractive. In this case, "close enough" is not a stringent criterion at all—to demonstrate this, start with any real number and repeatedly press the cos key on a calculator (checking first that the calculator is in "radians" mode). It eventually converges to about 0.739085133, which is a fixed point. That is where the graph of the cosine function intersects the line .

Not all fixed points are attractive: for example, x = 0 is a fixed point of the function f(x) = 2x, but iteration of this function for any value other than zero rapidly diverges. However, if the function f is continuously differentiable in an open neighbourhood of a fixed point x₀, and, attraction is guaranteed.

Attractive fixed points are a special case of a wider mathematical concept of attractors.

An attractive fixed point is said to be a stable fixed point if it is also Lyapunov stable.

A fixed point is said to be a neutrally stable fixed point if it is Lyapunov stable but not attracting. The center of a linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point.