According to Wolfram, a quadratic recurrence relation uses a second degree polynomial to express $x_{n+1}$ as a function of $x_n$. A "quadratic map", then, is a recurrence relation:
$$x_{n+1} = a (x_n)^2 + b x_n + c$$
Aside from Wolfram's Quadratic Map page, I can't seem to find much information on "Quadratic Maps". Is there some definitive reference on the topic, or some resource/reference that explains a lot of what is known about QMs?
Perhaps something like "Iterated Maps On The Interval As Dynamical Systems" by Collet and Eckmann . Additional: http://www.amazon.com/Iterated-Interval-Dynamical-Birkh%C3%A4user-Classics/dp/0817649263#reader_0817649263
Perhaps also Mira-Gumowski maps, these are generally real valued but perhaps not as simple as a quadratic map. I suggest you google the topic and view the images. Maybe that would be of interest.