Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom (Q379462)

The author considers nearly integrable Hamiltonians of the form NEWLINE\[NEWLINEh(x,I,\tau,\delta)= h_0(I)+\delta h_1(x,I,\tau,\delta),NEWLINE\]NEWLINE where \(\delta\ll 1\) is a small parameter, \((x,\tau)\in \mathbb{T}^2\), \(I\in\mathbb{R}\), and both \(h_0\) and \(h_1\) are analytic.NEWLINENEWLINE The subject of particular interest here is the dynamics in the region in which invariant tori exist for the unperturbed \((\delta=0)\) system but break down under perturbation. These resonance regions appear where the frequency vector \(\omega(I^*)\) of the unperturbed system is rationally dependent; this happens for \(I=I^*\) when \(\partial_I h_0(I^*)\) is rational. Under certain nondegeneracy conditions these resonances appear as lower-dimensional tori that are -- for systems with one and a half degrees of freedom--hyperbolic periodic orbits with stable and unstable manifolds.NEWLINENEWLINE The author's primary goal is to study possible transversal intersection of these invariant manifolds and the related chaotic dynamics. Standard Melnikov theory is not applicable here in this singular perturbation setting. The transversal intersection is exponentially small, and an asymptotic formula for the measure of the splitting is provided. To first-order this is \(Ke^\beta e^{-\alpha/\varepsilon}\). The author relates the constants \(K\), \(\beta\) and \(\alpha\) to system features.