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:
“Age becomes reality when you hear someone refer to “that attractive young woman standing next to the woman in the green dress,” and you find that you’re the one in the green dress.”
—Lois Wyse (20th century)
“I was not at all shocked with this execution at the time. John died seemingly without much pain. He was effectually hanged, the rope having fixed upon his neck very firmly, and he was allowed to hang near three quarters of an hour; so that any attempt to recover him would have been in vain. I comforted myself in thinking that by giving up the scheme I had avoided much anxiety and uneasiness.”
—James Boswell (1740–1795)
“Type of the wise, who soar, but never roam—
True to the kindred points of Heaven and Home!”
—William Wordsworth (1770–1850)