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- W2025746805 abstract "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 ensuremath{infty} (respectively, 0) after (respectively, before) the breaking point, it is numerically very difficult to estimate a critical residue value ${R}_{mathrm{cr}}$ for this cubic case (0.15ensuremath{lesssim}${R}_{mathrm{cr}ensuremath{lesssim}0.26}$). 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 (ensuremath{simeq}0.25) the parameter values ${k}_{n}$ do accumulate on a critical value, ${k}_{ensuremath{infty}}$, for torus breakup like ensuremath{Vert}${k}_{n}$-${k}_{nmathrm{ensuremath{-}}1}$ensuremath{Vert} ensuremath{sim}ensuremath{Vert}${p}_{n}$/${q}_{n}$-${p}_{nmathrm{ensuremath{-}}1}$/${q}_{nmathrm{ensuremath{-}}1}$${ensuremath{Vert}}^{ensuremath{xi}}$ as nensuremath{rightarrow}ensuremath{infty}, with a ``new'' scaling constant ensuremath{xi}ensuremath{approxeq}0.72, where ${p}_{n}$/${q}_{n}$ is 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 ensuremath{sim}${ensuremath{delta}}^{mathrm{ensuremath{-}}n}$." @default.
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- W2025746805 date "1989-01-01" @default.
- W2025746805 modified "2023-09-23" @default.
- W2025746805 title "Breakup of Kol’mogorov-Arnol’d-Moser tori of cubic irrational winding number" @default.
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- W2025746805 doi "https://doi.org/10.1103/physreva.39.344" @default.
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