### Abstract

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rssmann's nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.

Original language | English |
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Pages (from-to) | 3151-3161 |

Number of pages | 11 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 73 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2010 Nov 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- KAM theory
- Nearly integrable Hamiltonian systems
- Quasi-periodic motions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**A point-wise criterion for quasi-periodic motions in the KAM theory.** / Soga, Kohei.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 73, no. 10, pp. 3151-3161. https://doi.org/10.1016/j.na.2010.06.058

}

TY - JOUR

T1 - A point-wise criterion for quasi-periodic motions in the KAM theory

AU - Soga, Kohei

PY - 2010/11/15

Y1 - 2010/11/15

N2 - We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rssmann's nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.

AB - We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rssmann's nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.

KW - KAM theory

KW - Nearly integrable Hamiltonian systems

KW - Quasi-periodic motions

UR - http://www.scopus.com/inward/record.url?scp=77956179555&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956179555&partnerID=8YFLogxK

U2 - 10.1016/j.na.2010.06.058

DO - 10.1016/j.na.2010.06.058

M3 - Article

AN - SCOPUS:77956179555

VL - 73

SP - 3151

EP - 3161

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 10

ER -