  Explicit computational formulas for coefficients of the periodic normal forms
of the three most complex codim 2 bifurcations of limit cycles with dimension
of the center manifold equal to 4 or to 5 in generic autonomous ODEs are
derived. The resulting formulas are independent of the dimension of the phase
space and involve solutions of certain boundary-value problems as well as
multilinear functions from the Taylor expansion of the ODE right-hand side near
the cycle. The formulas allow one to distinguish between the complicated
bifurcation scenarios which can happen near these codim 2 bifurcations of limit
cycles, where 3-tori and 4-tori can be present. We apply our techniques to the
study of a known laser model, a novel model from population biology, and one
for mechanical vibrations; these models exhibit Limit Point--Neimark-Sacker,
Period-Doubling--Neimark-Sacker and double Neimark-Sacker bifurcations.
Lyapunov exponents are computed to numerically confirm the results of the
normal form analysis, in particular with respect to the existence of stable
invariant tori of various dimensions and chaos.
