The hyperbolic invariant tori of symplectic mappings

Zhu, Wenzhuang; Liu, Baifeng; Liu, Zhenxin

doi:10.1016/j.na.2006.10.035

Cited by 12 publications

(8 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, we note that the method of [48,66] leads to very good regularity conclusions for finite differentiable systems and also to good estimates on the measure occupied by the tori. We also call attention to [19,21,37,45,46,64,65,75] which consider also tori with hyperbolic and elliptic directions and relax the twist conditions and the differentiability requirements. The paper [26] considers analytic perturbations which depend only on the angles of reducible tori satisfying a twist condition and uses a direct resummation method.…”

Section: Introductionmentioning

confidence: 99%

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

Fontich

Llave

Sire³

2009

Journal of Differential Equations

126

View full text Add to dashboard Cite

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an a posteriori approach. The proof of the main theorems is based on an iterative method to solve the functional equation. The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map. The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on successive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.

show abstract

Section: Introductionmentioning

confidence: 99%

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

Fontich

Llave

Sire³

2009

Journal of Differential Equations

126

View full text Add to dashboard Cite

show abstract

“…In 2015, Lu et al [18] proved a KAM theorem on lower dimensional elliptic invariant tori for nearly integrable symplectic mappings. We refer readers to [19][20][21] for more results on symplectic mappings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A KAM Theorem for Lower Dimensional Elliptic Invariant Tori of Nearly Integrable Symplectic Mappings

Jiang¹

2017

Journal of Function Spaces

View full text Add to dashboard Cite

show abstract

“…More recently, Shang [5] considered the symplectic mappings with the highest dimensional tori, which is an analogy result of hamiltonian system [10]. Zhu [6] obtained persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings under a Rüssmann non-degeneracy condition.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the paper [6], the authors proved that most of lower hyperbolic dimensional tori can persist for small perturbation under a Rüssmann-type non-degenerate condition and some hypothesis. This result is an analogy of hamiltonian systems as in [7] and the proof is also similar to the hamiltonian case.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Since the method for hamiltonian systems depends strongly on the normal form for the integrable part, so we first need to know the normal form for symplectic mapping. For lower dimensional hyperbolic invariant tori as considered in the paper [6,7], it is easy to get the result since no resonant relation happens with the eigenvalues along normal direction.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

See 1 more Smart Citation

Persistence of Lower Dimensional Hyperbolic Invariant Tori for Nearly Integrable Symplectic Mappings

Bi¹,

2014

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

In this paper we consider a class of nearly integrable symplectic mappings with generating function and prove the persistence of lower dimensional hyperbolic invariant tori. Under a Rüssmann-type non-degenerate condition, by using the KAM theory, we proved that the nearly integrable twist symplectic mappings admit a family of lower dimensional invariant tori as long as the generating function is real analytic and the perturbation is sufficiently small.

show abstract

The hyperbolic invariant tori of symplectic mappings

Cited by 12 publications

References 14 publications

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

A KAM Theorem for Lower Dimensional Elliptic Invariant Tori of Nearly Integrable Symplectic Mappings

Persistence of Lower Dimensional Hyperbolic Invariant Tori for Nearly Integrable Symplectic Mappings

Contact Info

Product

Resources

About