Tinkerbell

Tinkerbell map — a quadratic map of the plane whose orbit sweeps a curved, wing-like region. Bounded only for a narrow band of parameters; push the knobs and it escapes to infinity, which the renderer catches and reseeds rather than drawing NaNs.

x' = x² − y² + a·x + b·y
y' = 2·x·y + c·x + d·y

It opens running, on the model selector, with its own knobs. Rotate it by dragging, zoom with the wheel, and turn the parameters to see what the figure does when the system changes.