We sketch the outlines of a perturbation scheme for the kick‐lattice map in two degrees of freedom (4‐d phase space) and apply it to the weak‐strong beam‐beam interaction. Our approach yields rigorous error bounds on the precision of our approximations (proofs will appear elsewhere), divides tune space into distinct regions where different approximations are valid, clarifies the dynamics near certain generically nonintegrable resonances, and generalizes to higher degrees of freedom and more complex models such as the strong‐strong beam‐beam. © 2003 American Institute of Physics