Editing Quadratic function (section)

==Iteration==
To [[iterated function|iterate a function]] <math>f(x)=ax^2+bx+c</math>, one applies the function repeatedly, using the output from one iteration as the input to the next.

One cannot always deduce the analytic form of <math>f^{(n)}(x)</math>, which means the ''n''<sup>th</sup> iteration of <math>f(x)</math>. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of <math>f(x)</math> if the inverse exists.) But there are some analytically [[closed-form expression|tractable]] cases.

For example, for the iterative equation

:<math>f(x)=a(x-c)^2+c</math>

one has
:<math>f(x)=a(x-c)^2+c=h^{(-1)}(g(h(x))),</math>
where
:<math>g(x)=ax^2</math> and <math>h(x)=x-c.</math>
So by induction,
:<math>f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))</math>
can be obtained, where <math>g^{(n)}(x)</math> can be easily computed as
:<math>g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.</math>
Finally, we have
:<math>f^{(n)}(x)=a^{2^n-1}(x-c)^{2^n}+c</math>

as the solution.

See [[Topological conjugacy]] for more detail about the relationship between ''f'' and ''g''. And see [[Complex quadratic polynomial]] for the chaotic behavior in the general iteration.

The [[logistic map]]

:<math> x_{n+1} = r x_n (1-x_n), \quad  0\leq x_0<1</math>

with parameter 2<''r''<4 can be solved in certain cases, one of which is [[chaos (mathematics)|chaotic]] and one of which is not. In the chaotic case ''r''=4 the solution is

:<math>x_{n}=\sin^{2}(2^{n} \theta \pi)</math>

where the initial condition parameter <math>\theta</math> is given by <math>\theta = \tfrac{1}{\pi}\sin^{-1}(x_0^{1/2})</math>.  For rational <math>\theta</math>, after a finite number of iterations <math>x_n</math> maps into a periodic sequence.  But almost all <math>\theta</math> are irrational, and, for irrational <math>\theta</math>, <math>x_n</math> never repeats itself – it is non-periodic and exhibits [[sensitive dependence on initial conditions]], so it is said to be chaotic.

The solution of the logistic map when ''r''=2 is

<math>x_n = \frac{1}{2} - \frac{1}{2}(1-2x_0)^{2^{n}}</math>

for <math>x_0 \in [0,1)</math>.  Since <math>(1-2x_0)\in (-1,1)</math> for any value of <math>x_0</math> other than the unstable fixed point 0, the term <math>(1-2x_0)^{2^{n}}</math> goes to 0 as ''n'' goes to infinity, so <math>x_n</math> goes to the stable fixed point <math>\tfrac{1}{2}.</math>