Exponentially small splitting of separatrices under fast quasiperiodic forcing
The authors consider a Hamiltonian system describing a quasiperiodic high-frequency perturbation of the pendulum and having a Hamiltonian of the form \[ \langle\omega,I\rangle/ \varepsilon+ h(x,y,\theta,\varepsilon), \] where \[ h=y^2/2+\cos x+\varepsilon^pm(\theta_1,\theta_2)\cos x, \] and the symplectic form is \(dx\wedge dy+d\theta_1\wedge dI_1+d\theta_2\wedge dI_2\). It is assumed that \(\varepsilon>0\) is a small parameter, \(p\geq 3\), \( \omega=(1,\gamma)\text{ with }\gamma=(\sqrt{5}+1)/2, \) and the function \(m\) is \(2\pi\)-periodic in \(\theta_1\) and \(\theta_2\). The authors investigate the splitting of invariant manifolds associated with a two-dimensional invariant torus. The main result estimates the difference between the splitting (exponentially small in \(\varepsilon\)) and the corresponding Melnikov function.
quasiperiodic high-frequency perturbation
scientific article; zbMATH DE number 1121188