On the break–down threshold of invariant tori in four dimensional maps Alessandra Celletti Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) (celletti@mat.uniroma2.it) Corrado Falcolini Dipartimento di Matematica Universit` a di Roma Tre Largo S. L. Murialdo 1, I-00146 Roma (Italy) (falco@mat.uniroma3.it) Ugo Locatelli Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) (locatell@mat.uniroma2.it) September 27, 2004 Abstract We investigate the break–down of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various two–dimensional frequency vectors, having eventually one or both components close to a rational value. The dynamics of this model is very reach and depends on two parameters, the perturbing and coupling parameters. Several techniques are introduced to determine the analyticity domain (in the complex perturbing parameter plane) and to compute the break–down threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Pad´ e approximants. The break–down threshold is computed through a suitable extension of Greene’s method to four dimensional systems. Frequency analysis is implemented and compared with the previous techniques. Keywords: Invariant tori, Greene’s method, Pad´ e approximants, Frequency analysis 1 Introduction The mechanism of break–down of invariant surfaces in nearly–integrable (continuous and dis- crete) systems has been widely studied through theoretical investigations and numerical ex- periments. In particular, the Kolmogorov–Arnold–Moser theory (see [17], [1] and [29]) allows to prove the existence of an invariant torus provided the perturbing parameter is sufficiently small. Many different techniques have been developed to compute the experimental value of the 1