Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems (Q1851411)

The authors consider Hamiltonians of the form \(H_{\mu}=w\cdot I+\frac{p_2}2+(\cos q-1)+\mu f(\varphi,q)\) with angle variables \((\varphi,q)\in T_n\times T_1\) and action variables \((I,p)\in\mathbb{R}_n\times\mathbb{R}_1\). This Hamiltonian describes a system of \(n\) isochronous harmonic oscillators of frequencies \(w\). When \(\mu=0\), the energy \(w_i I_i\) of each oscillator is a constant of the motion. When \(\mu\neq 0\), the question is whether there exist motions whose net effect is to transfer energy from one oscillator to the others via Arnold diffusion. The authors make the assumption (H1) that there exists \(\gamma>0\) and \(\tau>n\) such that \(|w\cdot k|\geq\gamma /|k|^\tau\) for all \(k\in\mathbb{Z}_n\) with \(k\neq 0\). For \(\mu=0\), the Hamiltonian \(H_{\mu}\) admits a family of invariant \(n\)-dimensional tori. When \(\mu\neq 0\), the authors' hypothesis H1 leads to the persistence of the invariant tori when \(\mu\) is sufficiently small. The authors prove a shadowing lemma that provides an improved estimate for the Arnold diffusion time. In addition, they demonstrate for three time scales that the Poincaré-Melnikov function correctly predicts the splitting of the system separatrices. Complete proofs and additional results are provided in the authors' preprint [\textit{M. Berti} and \textit{P. Bolle}, Arnold's diffusion for nearly integrable isochronous Hamiltonian systems, Preprint SISSA 98/2000/M, October 2000].