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

 (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
 (2)

 (3)

 (4)

Period two Fixed Points occur when
 (5)

 (6)

 (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

 (9) (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
 (11) (12)

The inverse map is
 (13) (14)

The Fixed Points are given by
 (15)

for , ..., .

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

© 1996-9 Eric W. Weisstein
1999-05-25