The Henon map takes the vector [x y] to [x, y - x*x] and then rotates it through an angle A. It is an area-preserving map, and simulates the Poincare map of period orbits in Hamiltonian systems.
The following applet plots the orbits under the Henon map of the points selected by the mouse. It toggles between "Plot" and "Zoom" mode. The angle is selectable, as is the number of points in each orbit we plot. It implements zooming, and the zoom factor is also selectable.