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Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems
- Source :
- Science China Mathematics. 61:1567-1588
- Publication Year :
- 2018
- Publisher :
- Springer Science and Business Media LLC, 2018.
-
Abstract
- In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying R\"{u}ssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus, the one-step map of the scheme is conjugate to a one parameter family of linear rotations with a step size dependent frequency vector in terms of iteration. These results are a generalization of Shang's theorems (1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov. In comparison, R\"{u}ssmann's condition is the weakest non-degeneracy condition for the persistence of invariant tori in Hamiltonian systems. These results provide new insight into the nonlinear stability of symplectic integrators.<br />Comment: It has been accepted for publication in SCIENCE CHINA Mathematics
- Subjects :
- Pure mathematics
Integrable system
Kolmogorov–Arnold–Moser theorem
General Mathematics
Diophantine equation
010102 general mathematics
37J35, 37J40, 65L07, 65L20, 65P10, 65P40
Torus
Dynamical Systems (math.DS)
Numerical Analysis (math.NA)
01 natural sciences
Hamiltonian system
010101 applied mathematics
Mathematics - Classical Analysis and ODEs
Phase space
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
Mathematics - Numerical Analysis
Mathematics - Dynamical Systems
0101 mathematics
Invariant (mathematics)
Mathematics::Symplectic Geometry
Symplectic geometry
Mathematics
Subjects
Details
- ISSN :
- 18691862 and 16747283
- Volume :
- 61
- Database :
- OpenAIRE
- Journal :
- Science China Mathematics
- Accession number :
- edsair.doi.dedup.....2c9e5d7ecf2a4d89d4a86d97d1880da8