A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters (Q2758134)

The authors study the following Hamiltonian: NEWLINE\[NEWLINEH(x,I_2, I_3,y, \theta_2,\theta_3) =\frac 12(x^2+ I^2_2)+I_3+ \varepsilon (\cos y-1) \bigl( 1+\mu(\cos \theta_2+ \cos\theta_3) \bigr). \tag{1}NEWLINE\]NEWLINE This is a classical examples displaying a (very slow) drift of the actions for small values of \(\varepsilon\), provided \(\mu\) is small enough. Here the authors take \(\varepsilon =\mu\) and present formal approximations of the three-dimensional invariant manifolds associated with this torus and numerical globalization of these manifolds. This allows to obtain the splitting (of separatrices) vector and to compute its Fourier components. Note that the main ideas of this paper, concerning evaluation of the splitting are as follows:NEWLINENEWLINENEWLINE(a) instead of looking for estimates of the splitting at once, they look for the contributions of the different harmonics to the splitting individually,NEWLINENEWLINENEWLINE(b) a given harmonic can be not present in the initial Hamiltonian, it can appear when some averaging steps are carried out.