Oscillatory Escape in a Duffing Equation with a Polynomial Potential

We show that the time-periodic Hamiltonian systemsd2x/dt2+x2n+1+a(t)x2l+1=0, 2n>2l>n, with a discontinuity ina(t), possess unbounded solutionsx(t) which, moreover, oscillate between a finite disk and infinity; in particular lim inft→∞ x(t)<∞ and lim supt→∞ x(t)=∞. As a consequence, the Poincaré map possesses no invariant KAM curves enclosing the origin outside a bounded disk.