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The elliptical invariant tori of nearly integrable Hamiltonian system through symplectic algorithms

Zaijiu Shang, Yang Xu

Abstract

In this paper we apply symplectic algorithms to nearly integrable Hamiltonian system, and prove it can maintain lots of elliptic lower dimensional invariant tori. We are committed to consider the elliptic lower dimensional invariant tori for symplectic mapping with a small twist under the Rüssmann's non-degenerate condition, and focus on its measure estimation. And then apply it to the nearly integrable Hamiltonian system to obtain lots of elliptic lower dimensional invariant tori.

The elliptical invariant tori of nearly integrable Hamiltonian system through symplectic algorithms

Abstract

In this paper we apply symplectic algorithms to nearly integrable Hamiltonian system, and prove it can maintain lots of elliptic lower dimensional invariant tori. We are committed to consider the elliptic lower dimensional invariant tori for symplectic mapping with a small twist under the Rüssmann's non-degenerate condition, and focus on its measure estimation. And then apply it to the nearly integrable Hamiltonian system to obtain lots of elliptic lower dimensional invariant tori.
Paper Structure (11 sections, 10 theorems, 110 equations)

This paper contains 11 sections, 10 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preprocessing
  3. Main Conclusions
  4. The Symplectic Mapping System with a Small Twist (\ref{['e01']})
  5. Measure estimation
  6. Proof of Theorem \ref{['t1']}
  7. Proof of Theorem \ref{['t2']}
  8. Proof
  9. Applied to the symplectic difference scheme (\ref{['004']})
  10. Proof of Corollary \ref{['co1']}
  11. Proof of Corollary \ref{['co2']}

Key Result

Theorem 3.1

For the symplectic mapping $F$ defined by (e01) , $t H = t N + t P$, assume $h$ and $P$ are real analytic on a complex neighborhood $D$ of $\mathbb{T}^n \times W \times V \times W$, $\omega$ satisfies the Rüssmann's non-degeneracy condition and (001), and (e04) holds, $\sup_{D} |P(x,u,\hat{y},\hat{v

Theorems & Definitions (17)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.1
  • Corollary 3.2
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 5.1
  • Lemma 5.1
  • ...and 7 more