Destruction of Invariant Tori in Pendulum-Type Equations

In this paper we prove that if there exists an invariant torus with the rotation number (1, ω) in the pendulum-type equationx=Q0 x(t, x) for a given potentialQ0=Q0(t, x) C∞(T2), andωis a Liouville number, then for any neighborhood (Q0) ofQ0in theC∞topology, there exists a potentialQ=Q(t, x) (Q0) such that the systemx=Qx(t, x) does not admit any invariant torus with the rotation number (1, ω). This confirms J. Moserʹs suggestion inBol. Soc. Brasil. Mat.20(1981), 29–45.