Editing Attractor (section)

=== Fixed point ===
[[File:Critical orbit 3d.png|right|thumb|Weakly attracting fixed point for a complex number evolving according to a [[complex quadratic polynomial]]. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor.]]
A [[Fixed point (mathematics)|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 corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a [[damping ratio|damped]] [[pendulum]], the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between [[Stability theory#Stability of fixed points|stable and unstable equilibria]]. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium).

In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the [[nonlinear dynamics]] of [[stiction]], [[friction]], [[surface roughness]], [[Deformation (engineering)|deformation]] (both [[Elastic deformation|elastic]] and [[plastic]]ity), and even [[quantum mechanics]].<ref name="Contact of Nominally Flat Surfaces">{{cite journal|last=Greenwood|first=J. A.|author2=J. B. P. Williamson|title=Contact of Nominally Flat Surfaces|journal=Proceedings of the Royal Society|date=6 December 1966|volume=295|issue=1442|pages=300–319|doi=10.1098/rspa.1966.0242|bibcode=1966RSPSA.295..300G |s2cid=137430238}}</ref> In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly [[Sphere#Hemisphere|hemispherical]], and the marble's [[sphere|spherical]] shape, are both much more complex surfaces when examined under a microscope, and their [[Contact mechanics#History|shapes change]] or [[Deformation (physics)|deform]] during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains.<ref name="NISTIR 89-4088">{{cite book|last=Vorberger|first=T. V.|title=Surface Finish Metrology Tutorial|year=1990|publisher=U.S. Department of Commerce, National Institute of Standards (NIST)|page=5|url=https://www.nist.gov/calibrations/upload/89-4088.pdf}}</ref> There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered [[Critical point (mathematics)|stationary]] or fixed points, some of which are categorized as attractors.