Editing Chaos theory (section)

===Strange attractors===
[[File:TwoLorenzOrbits.jpg|thumb|right|The [[Lorenz attractor]] displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.]]

Some dynamical systems, like the one-dimensional [[logistic map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x''),</span> are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an [[attractor]], since then a large set of initial conditions leads to orbits that converge to this chaotic region.<ref>{{cite journal|last1=Strelioff|first1=Christopher|last2=et.|first2=al.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|pages=044101|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|bibcode = 2006PhRvL..96d4101S }}</ref>

An easy way to visualize a chaotic attractor is to start with a point in the [[basin of attraction]] of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the [[Edward Lorenz|Lorenz]] weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly.

Unlike [[Attractor#Fixed point|fixed-point attractors]] and [[limit cycle]]s, the attractors that arise from chaotic systems, known as [[strange attractor]]s, have great detail and complexity. Strange attractors occur in both [[continuous function|continuous]] dynamical systems (such as the Lorenz system) and in some [[discrete mathematics|discrete]] systems (such as the [[Hénon map]]). Other discrete dynamical systems have a repelling structure called a [[Julia set]], which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a [[fractal]] structure, and the [[fractal dimension]] can be calculated for them.