Editing Chaos theory (section)

===Topological mixing===
[[File:LogisticTopMixing1-6.gif|thumb|Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that ''mixing'' occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4  x_k  (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.]]
[[File:Chaos Topological Mixing.png|thumb|The map defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> also displays [[topological mixing]]. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space.]]

[[Topological mixing]] (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or [[open set]] of its [[phase space]] eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored [[dye]]s or fluids is an example of a chaotic system.

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.