  In this paper, we show that the destruction of the main KAM islands in
two-degree-of-freedom Hamiltonian systems occurs through a cascade of
period-doubling bifurcations. We calculate the corresponding Feigenbaum
constant and the accumulation point of the period-doubling sequence. By means
of a systematic grid search on exit basin diagrams, we find the existence of
numerous very small KAM islands ('islets') for values below and above the
aforementioned accumulation point. We study the bifurcations involving the
formation of islets and we classify them in three different types. Finally, we
show that the same types of islets appear in generic two-degree-of-freedom
Hamiltonian systems and in area-preserving maps.
