Approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom

We construct an approximate renormalization transformation that combines Ko lmogorov-Arnol'd-Moser and renormalization-group techniques, to analyze ins tabilities in Hamiltonian systems with three degrees of freedom. This schem e is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically th at the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetica lly degenerate and nondegenerate Hamiltonians belong to the same universali ty class, and thus the corresponding critical invariant tori have the same type of scaling properties. We numerically investigate the structure of the attracting set on the critical surface and find that it is a strange nonch aotic attractor. We compute exponents that characterize its universality cl ass. [S1063-651X(99)04211-7].