Heinz Hanssman (Mathematisch Instituut, Universiteit Utrecht (Holanda)): On the destruction of resonant Lagrangean tori in Hamiltonian Systems.

Poincaré's fundamental problem of dynamics concerns the behaviour
of an integrable Hamiltonian system under a (small) non-integrable
perturbation. Under rather weak conditions K(olmogorov)A(rnol'd)M(oser)
theory settles this question for the majority of initial values. The
perturbed motion is (again) quasi-periodic, the number of frequencies
equals the number of degrees of freedom. KAM theory proves such
Lagrangean tori to persist provided that the frequencies are bounded
away from resonances by means of Diophantine inequalities.

How do Lagrangean tori with resonant frequencies behave under
perturbation ? We concentrate on a single resonance, whence many
n-parameter families of n-tori are expected to be generated by the
perturbation; here n+1 is the number of degrees of freedom. For
non-degenerate systems we explain the pattern how these families of
lower-dimensional tori come into existence, and then discuss what
happens in the presence of degeneracies.