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Quadratic Map

A 1-D Map often called ``the'' quadratic map is defined by

x_{n+1} ={x_n}^2+c.
\end{displaymath} (1)

This is the real version of the complex map defining the Mandelbrot Set. The quadratic map is called attracting if the Jacobian $J<1$, and repelling if $J>1$. Fixed Points occur when
x^{(1)} = [x^{(1)}]^2+c
\end{displaymath} (2)

\end{displaymath} (3)

x^{(1)}_\pm = {\textstyle{1\over 2}}(1\pm \sqrt{1-4c}\,).
\end{displaymath} (4)

Period two Fixed Points occur when
$\displaystyle x_{n+2}$ $\textstyle =$ $\displaystyle {x_{n+1}}^2+c=({x_n}^2+c)^2+c$  
  $\textstyle =$ $\displaystyle {x_n}^4+2c{x_n}^2+(c^2+c)=x_n$ (5)

\end{displaymath} (6)

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} (7)

Period three Fixed Points occur when
$+(c+1)^2x+(c^3+2c^2+c+1)=0.\quad$ (8)

The most general second-order 2-D Map with an elliptic fixed point at the origin has the form

$\displaystyle x'$ $\textstyle =$ $\displaystyle x\cos\alpha-y\sin\alpha+a_{20}x^2+a_{11}xy+a_{02}y^2$ (9)
$\displaystyle y'$ $\textstyle =$ $\displaystyle x\sin\alpha+y\cos\alpha+b_{20}x^2+b_{11}xy+b_{02}y^2.$ (10)

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
$\displaystyle x'$ $\textstyle =$ $\displaystyle x\cos\alpha-y\sin\alpha+x^2\sin\alpha$ (11)
$\displaystyle y'$ $\textstyle =$ $\displaystyle x\sin\alpha+y\cos\alpha-x^2\cos\alpha.$ (12)

The inverse map is
$\displaystyle x$ $\textstyle =$ $\displaystyle x'\cos\alpha+y'\sin\alpha$ (13)
$\displaystyle y$ $\textstyle =$ $\displaystyle -x'\sin\alpha+y'\cos\alpha+(x'\cos\alpha+y'\sin\alpha)^2.$ (14)

The Fixed Points are given by
{x_i}^2\sin\alpha+2x_i\cos\alpha-x_{i-1}-x_{i+1} = 0
\end{displaymath} (15)

for $i = 0$, ..., $n-1$.

See also Bogdanov Map, Hénon Map, Logistic Map, Lozi Map, Mandelbrot Set

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© 1996-9 Eric W. Weisstein