Chaos

Q3: What is chaos?
A3: Chaos is apparently unpredictable behavior arising in a deterministic sys-
tem because of great sensitivity to initial conditions.  Chaos arises in a
dynamical system if two arbitrarily close starting points diverge exponential-
ly, so that their future behavior is eventually unpredictable.

Weather is considered chaotic since arbitrarily small variations in initial
conditions can result in radically different weather later.  This may limit
the possibilities of long-term weather forecasting.  (The canonical example is
the possibility of a butterfly's sneeze affecting the weather enough to cause
a hurricane weeks later.)

Devaney defines a function as chaotic if it has sensitive dependence on ini-
tial conditions, it is topologically transitive, and periodic points are
dense.  In other words, it is unpredictable, indecomposable, and yet contains
regularity.

Allgood and Yorke define chaos as a trajectory that is exponentially unstable
and neither periodic or asymptotically periodic.  That is, it oscillates ir-
regularly without settling down.