  Periodic normal forms for the codim 2 bifurcations of limit cycles up to a
3-dimensional center manifold in generic autonomous ODEs and computational
formulas for their coefficients are derived. The formulas are independent of
the dimension of the phase space and involve solutions of certain
boundary-value problems on the interval [0, T ], where T is the period of the
critical cycle, as well as multilinear functions from the Taylor expansion of
the right-hand sides near the cycle. The formulas allow us to distinguish
between various bifurcation scenarios near codim 2 bifurcations. Our
formulation makes it possible to use robust numerical boundary-value algorithms
based on orthogonal collocation, rather than shooting techniques, which greatly
expands its applicability. The actual implementation is described in detail
with numerical examples.
