Breakup of Kol’mogorov-Arnol’d-Moser tori of cubic irrational winding number

For the special case in which the irrational winding number is the root of a cubic equation, we present numerical evidence for the validity of some form of residue criterion [Greene, J. Math. Phys. 20, 1182 (1979)]. This is a criterion for the breakup of Kol'mogorov-Arnol'd-Moser tori in two degrees of freedom. The cubic case is essential for future work on four-dimensional maps. While the residues do approach {∞} (respectively, 0) after (respectively, before) the breaking point, it is numerically very difficult to estimate a critical residue value Rcr for this cubic case (0.15{}R_cr0.26). We do not find the usual ``scaling'' (Greene, ibid., and Mackay, Ph.D. thesis, Princeton University, 1982). For one particular cubic irrational, however, we have obtained a different form of scaling: At constant residue ({}0.25) the parameter values kₙ do accumulate on a critical value, k_∞, for torus breakup like {}kₙ-k_n-1{} {~}{}pₙ/qₙ-p_n-1/q_n-1{{}}^{{ξ}}as n→∞, with a ``new'' scaling constant ξ0.72, where{p}ₙ$/${q}ₙis the nth rational approximant in the continued-fraction expansion of the cubic irrational. For a quadratic irrational this scaling reduces to the usual power-law scaling ~{{δ}}^{{{-}}n}$.