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Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off. (English) Zbl 0933.37059

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.

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
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