  We construct a simple symplectic map to study the dynamics of eccentric
orbits in non-spherical potentials. The map offers a dramatic improvement in
speed over traditional integration methods, while accurately representing the
qualitative details of the dynamics. We focus attention on planar,
non-axisymmetric power-law potentials, in particular the logarithmic potential.
We confirm the presence of resonant orbit families (``boxlets'') in this
potential and uncover new dynamics such as the emergence of a stochastic web in
nearly axisymmetric logarithmic potentials. The map can also be applied to
triaxial, lopsided, non-power-law and rotating potentials.
