A 1-D Map often called ``the'' quadratic map is defined by
![\begin{displaymath}
x_{n+1} ={x_n}^2+c.
\end{displaymath}](q_410.gif) |
(1) |
This is the real version of the complex map defining the Mandelbrot Set. The quadratic map is called attracting if the
Jacobian
, and repelling if
. Fixed Points occur when
![\begin{displaymath}
x^{(1)} = [x^{(1)}]^2+c
\end{displaymath}](q_413.gif) |
(2) |
![\begin{displaymath}
(x^{(1)})^2-x^{(1)}+c=0
\end{displaymath}](q_414.gif) |
(3) |
![\begin{displaymath}
x^{(1)}_\pm = {\textstyle{1\over 2}}(1\pm \sqrt{1-4c}\,).
\end{displaymath}](q_415.gif) |
(4) |
Period two Fixed Points occur when
![\begin{displaymath}
x^4+2x^2-x+(cx^2+c)=(x^2-x+c)(x^2+x+1+c)=0
\end{displaymath}](q_419.gif) |
(6) |
![\begin{displaymath}
x^{(2)}_\pm = {\textstyle{1\over 2}}[1\pm\sqrt{1-4(1+c)}\,] = {\textstyle{1\over 2}}(1\pm\sqrt{-3-4c}\,).
\end{displaymath}](q_420.gif) |
(7) |
Period three Fixed Points occur when
|
|
|
(8) |
The most general second-order 2-D Map with an elliptic fixed point at the origin has the form
The map must have a Determinant of 1 in order to be Area preserving, reducing
the number of independent parameters from seven to three. The map can then be put in a
standard form by scaling and rotating to obtain
The inverse map is
The Fixed Points are given by
![\begin{displaymath}
{x_i}^2\sin\alpha+2x_i\cos\alpha-x_{i-1}-x_{i+1} = 0
\end{displaymath}](q_429.gif) |
(15) |
for
, ...,
.
See also Bogdanov Map, Hénon Map, Logistic Map,
Lozi Map, Mandelbrot Set
© 1996-9 Eric W. Weisstein
1999-05-25