A new method for measuring the splitting of invariant manifolds (Q2731038)

The main object of the paper is a Hamiltonian system, a generalisation of the classical Arnold example, described by the Hamiltonian function NEWLINE\[NEWLINE H=\omega\cdot I+\tfrac{\alpha}{2}I^ 2+\tfrac{1}{2} p^2+ \varepsilon\cos q+\varepsilon\mu F(q,\varphi). NEWLINE\]NEWLINE Here \(q\in{\mathbb T}\) and \(\varphi\in{\mathbb T}^d\) are angles; \(p\in{\mathbb R}\) and \(I\in{\mathbb R}^d\) are corresponding actions; \(\varepsilon\) and \(\mu\) are small parameters. The system has \(d+1\) degrees of freedom. If \(\mu=0\) the system is integrable and has partially hyperbolic invariant tori with doubled stable and unstable invariant manifolds. It is assumed that the perturbation vanishes on these tori. The invariant manifolds associated with the invariant tori may split under the perturbation.NEWLINENEWLINENEWLINEThe main result of the paper is an exponentially small upper bound for the splitting of these invariant manifolds. In the case of 3 degrees of freedom it is shown that the splitting of invariant manifolds is correctly predicted by the Melnikov method at least for a torus with the golden-mean ration of the frequencies.NEWLINENEWLINENEWLINEThe proofs are based on a delicate analysis of analytical continuations of solutions to the Hamilton-Jacobi equation.