ESTIMATE OF THE TRANSITION VALUE OF LIBRATIONAL INVARIANT CURVES ALESSANDRA CELLETTI 1 , GABRIELLA DELLA PENNA 2 and CLAUDE FROESCHL ´ E 2 1 Dipt. di Matematica, Universit´ a di Roma ‘Tor Vergata’ Via della Ricerca Scientifica – I-00133, Roma, Italy, e-mail: celletti@mat.uniroma2.it 2 Observatoire de Nice BP 229, F-06304 Nice Cedex 4, France, e-mails: gabry@obs-nice.fr; claude@obs-nice.fr Abstract. We investigate the break-down threshold of librational invariant curves. As a model problem, we consider a variant of a mapping introduced by M. H´ enon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene’s method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However, this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the fast Lyapunov indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene’s method for the computation of the critical parameter. Key words: librational curves, break-down threshold, fast Lyapunov indicators, H´ enon’s mapping 1. Introduction The study of two degrees of freedom Hamiltonian systems is often reduced to the investigation of area-preserving mappings by using a surface of section proced- ure. Indeed, such mappings provide many useful informations about the dynamical behaviour of the trajectories. A special class of area-preserving mappings is rep- resented by nearly-integrable maps, which depend upon a perturbing parameter, say ε. For ε = 0 the motion is integrable, while chaotic dynamics appear as soon as ε is not zero. We analyze the case when the mapping has an elliptic fixed point, which might be surrounded by librational curves. As in a pendulum- like structure, a chaotic separatrix divides the region of librational motion from the region where rotational invariant curves can be observed (see Figure 1). By KAM theory (Kolmogorov, 1954; Moser, 1962; Arnold, 1963), we know that the invariant curves (of either type librational or rotational) survive for small values of ε. For a given rotation number, say ω, of the invariant curve, there exists a critical value of the perturbing parameter, denoted by ε c (ω), at which the invariant curve disappears. A lower bound on ε c (ω) can be analytically obtained by KAM theory. The problem of the numerical evaluation of ε c (ω) was addressed by several authors Celestial Mechanics and Dynamical Astronomy 83: 257–274, 2002. c  2002 Kluwer Academic Publishers. Printed in the Netherlands.