  We study dynamics of a generic quadratic diffeomorphism, a 3D generalization
of the planar H\'{e}non map. Focusing on the dissipative, orientation
preserving case, we give a comprehensive parameter study of codimension-one and
two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker
bifurcations, give rise to Arnold tongues in parameter space. Aperiodic
attractors include invariant circles and chaotic orbits; these are
distinguished by rotation number and Lyapunov exponents. Chaotic orbits include
H\'{e}non-like and Lorenz-like attractors, which can arise from period-doubling
cascades, and those born from the destruction of invariant circles. The latter
lie on paraboloids near the local unstable manifold of a fixed point.
