INACCESSIBLE ATTRACTORS OF WEAKLY DISSIPATIVE SYSTEMS

We consider weakly dissipative perturbations of Hamiltonian systems of the form where V is a three times differentiable 2 pi-periodic functi on, with two maxima and two minima per period. The dissipation is para metrized by d > 0 and as d --> 0 the system tends to the Hamiltonian s ystem. With the addition of dissipation, the system develops a pair of attractors, B and D (module the periodicity in x), one near each of t he minima of V. We show that for typical systems these can be chosen s o that as d --> 0 there exists a countable set, J(k), of non-trivial d isjoint intervals accumulating on zero from above such that if d is an element of J(k) then no solutions with sufficiently large initial ene rgy and initial velocity of a given sign can be attracted to D (i.e. a lmost all such solutions tend to B, independent of the initial speed).