Lowenstein, John; Hatjispyros, Spyros; Vivaldi, Franco Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off. (English) Zbl 0933.37059 Chaos 7, No. 1, 49-66 (1997). Summary: We investigate the effects of round-off errors on the orbits of a linear symplectic map of the plane, with rational rotation number \(\nu= p/q\). Uniform discretization transforms this map into a permutation of the integer lattice \(\mathbb{Z}^2\). We study in detail the case \(q = 5\), exploiting the correspondence between \(\mathbb{Z}\) and a suitable domain of algebraic integers. We completely classify the orbits, proving that all of them are periodic. Using higher-dimensional embedding, we establish the quasi-periodicity of the phase portrait. We show that the model exhibits asymptotic scaling of the periodic orbits and a long-range clustering property similar to that found in repetitive tilings of the plane. Cited in 3 ReviewsCited in 22 Documents MSC: 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 37C27 Periodic orbits of vector fields and flows 65P10 Numerical methods for Hamiltonian systems including symplectic integrators 65G50 Roundoff error Keywords:linear symplectic map; rational rotation number; quasiperiodicity of the phase portrait; periodic orbits; repetitive tilings; round-off scheme PDFBibTeX XMLCite \textit{J. Lowenstein} et al., Chaos 7, No. 1, 49--66 (1997; Zbl 0933.37059) Full Text: DOI References: [1] DOI: 10.1080/10586458.1994.10504299 · Zbl 0832.58017 · doi:10.1080/10586458.1994.10504299 [2] Rannou F., Astron. Astrophys. 31 pp 289– (1974) [3] DOI: 10.1016/0167-2789(83)90232-4 · doi:10.1016/0167-2789(83)90232-4 [4] Kaneko K., Phys. Lett. A 129 pp 9– (1988) · doi:10.1016/0375-9601(88)90464-1 [5] Scovel C., Phys. Lett. A 159 pp 396– (1991) · doi:10.1016/0375-9601(91)90368-I [6] Earn D. J. D., Physica D 56 pp 1– (1992) · Zbl 0759.58015 · doi:10.1016/0167-2789(92)90047-Q [7] DOI: 10.1016/0167-2789(94)00103-0 · Zbl 0816.58026 · doi:10.1016/0167-2789(94)00103-0 [8] Lowenstein J. H., Phys. Rev. E 49 pp 231– (1994) · doi:10.1103/PhysRevE.49.232 [9] DOI: 10.1063/1.166126 · doi:10.1063/1.166126 [10] Zaslavskii G. M., Usp. Fiz. Nauk 156 pp 193– (1988) · doi:10.3367/UFNr.0156.198810a.0193 [11] Zaslavskii G. M., Sov. Phys. Usp. 31 pp 887– (1988) · doi:10.3367/UFNr.0156.198810a.0193 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.