Page "Attractor" Paragraph 23

A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation.

If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation.

The final state that a dynamical system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass, corresponds to an attracting fixed point of the evolution function, but the two concepts are not equivalent because not all fixed points attract the evolution of nearby points.

A marble rolling around in a basin may have a fixed point, but if the marble is externally driven it may not be attracted to that fixed point.

But in the absence of an external driving force, it will settle into the fixed point at the bottom of the bowl, so in this case that point is an attractor.