Quadratic Map

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

$$x_{n+1} ={x_n}^2+c.$$ (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

$$x^{(1)} = [x^{(1)}]^2+c$$ (2)

$$(x^{(1)})^2-x^{(1)}+c=0$$ (3)

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

Period two Fixed Points occur when

$$x_{n+2} = {x_{n+1}}^2+c=({x_n}^2+c)^2+c$$

$$= {x_n}^4+2c{x_n}^2+(c^2+c)=x_n$$ (5)

$$x^4+2x^2-x+(cx^2+c)=(x^2-x+c)(x^2+x+1+c)=0$$ (6)

$$x^{(2)}_\pm = {\textstyle{1\over 2}}[1\pm\sqrt{1-4(1+c)}\,] = {\textstyle{1\over 2}}(1\pm\sqrt{-3-4c}\,).$$ (7)

Period three Fixed Points occur when

$x^6+x^5+(3c+1)x^4+(2c+1)x^3+(c^2+3c+1)x^2$
$+(c+1)^2x+(c^3+2c^2+c+1)=0.\quad$	(8)

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

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

$$y' = x\sin\alpha+y\cos\alpha+b_{20}x^2+b_{11}xy+b_{02}y^2.$$ (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

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

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

The inverse map is

$$x = x'\cos\alpha+y'\sin\alpha$$ (13)

$$y = -x'\sin\alpha+y'\cos\alpha+(x'\cos\alpha+y'\sin\alpha)^2.$$ (14)

The Fixed Points are given by

$${x_i}^2\sin\alpha+2x_i\cos\alpha-x_{i-1}-x_{i+1} = 0$$ (15)

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

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