Chaos in Newton's Method
by Dan Kalman

Newton's method is a process for approximating roots of a function.  Usually the method produces a sequence that converges rapidly to a root.  However, viewed globally, Newton's method can exhibit chaotic behavior, as described in my book Uncommon Excursions.

Below there is a graphical dramatization of the following description of Newton's method.  From any point on the x axis: go vertically to the graph of the function; slide along the tangent line to the x axis; repeat.  Further down there are additional animations that concern chaos.

The next two animations illustrate the appearance in Newton's method of sensitive dependence on initial conditions. For one specific function, Newton's method is applied starting with two different initial values differing by a tiny amount, .00000000001. In spite of this miniscule difference, Newton's method for these two initial values converges to completely different roots of the given function. The animations also illustrate the possibility of cyclic patterns arising in Newton's method. In particular, the two initial values are both very close to a point of period six.