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.
Read more about this topic: Fixed Point (mathematics)
Famous quotes containing the words attractive, fixed and/or points:
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—Calder Willingham (1923–1995)
“At first I intended to become a student of the Senate rules and I did learn much about them, but I soon found that the Senate had but one fixed rule, subject to exceptions of course, which was to the effect that the Senate would do anything it wanted to do whenever it wanted to do it.”
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“The dominant metaphor of conceptual relativism, that of differing points of view, seems to betray an underlying paradox. Different points of view make sense, but only if there is a common co-ordinate system on which to plot them; yet the existence of a common system belies the claim of dramatic incomparability.”
—Donald Davidson (b. 1917)