On lower dimensional invariant tori in C d reversible systems

Zhang, Jing

doi:10.1007/s11401-008-0082-1

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction and Main Results1

Lu Chen1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2010

2024

Publication Types

Select...

Article5

Other2

Relationship

Self Cite0

Independent7

Authors

Journals

Cited by 9 publications

(2 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The KAM theory for analytic reversible equations (vector-fields) of more general form was deeply investigated in Sevryuk [30,31,32,33,34] and Broer [4]. Zhang [40] constructed a KAM theorem for a class of reversible equations which are assume to be C ℓ smooth where the low bound of ℓ < ∞ is not specified. Sevryuk [30] also studied deeply the KAM theory for reversible mappings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

KAM theorem for reversible mapping of low smoothness with application

Li¹,

Qi²,

Yuan³

2019

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

KAM theorem for reversible mapping of low smoothness with application

Li¹,

Qi²,

Yuan³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The KAM theory for reversible equations (vector-fields) of more general form was deeply investigated in [7][8][9][10][11][12][13][14][15]. In addition, there are many applications of reversible KAM theory (see [16][17][18][19][20]).…”

Section: Lu Chenmentioning

confidence: 99%

On invariant tori in some reversible systems

Chen¹

2020

Preprint

View full text Add to dashboard Cite

In the present paper, we consider the following reversible system ẋ = ω 0 + f (x, y), ẏ = g(x, y),) and f , g are reversible with respect to the involution G: (x, y) → (−x, y), that is, f (−x, y) = f (x, y), g(−x, y) = −g(x, y). We study the accumulation of an analytic invariant torus Γ 0 of the reversible system with Diophantine frequency ω 0 by other invariant tori. We will prove that if the Birkhoff normal form around Γ 0 is 0-degenerate, then Γ 0 is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive and the density of the union of these tori at Γ 0 being one. We will also prove that if the Birkhoff normal form around Γ 0 is j-degenerate (1 ≤ j ≤ d − 1) and condition (1.6) is satisfied, then through Γ 0 there passes an analytic subvariety of dimension d + j foliated into analytic invariant tori with frequency vector ω 0 . If the Birkhoff normal form around Γ 0 is d − 1-degenerate, we will prove a stronger result, that is, a full neighborhood of Γ 0 is foliated into analytic invariant tori with frequency vectors proportional to ω 0 .

show abstract