Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori

Gaidashev, Denis

doi:10.3934/dcds.2005.13.63

Cited by 18 publications

(24 citation statements)

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“…(This applies to an individual Hamiltonian satisfying a non-degeneracy condition, if one considers the family of its translates.) Our goal is to obtain such results within the framework of renormalization transformations [6,7,10,11,16,17,[19][20][21][22][23][25][26][27], and, more importantly, to develop appropriate techniques for analyzing quasiperiodic motion in a large class of flows. Our goal is to obtain such results within the framework of renormalization transformations [6,7,10,11,16,17,[19][20][21][22][23][25][26][27], and, more importantly, to develop appropriate techniques for analyzing quasiperiodic motion in a large class of flows.…”

Section: Introductionmentioning

confidence: 99%

Renormalization of vector fields and Diophantine invariant tori

Koch

Kocić

2008

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385707000892 How to cite this article: HANS KOCH and SAŠA KOCIĆ (2008). Renormalization of vector elds and Diophantine invariant tori.Abstract. We extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on T d × R . Each Diophantine vector ω ∈ R d determines an analytic manifold W of infinitely renormalizable vector fields, and each vector field on W is shown to have an elliptic invariant d-torus with frequencies ω 1 , ω 2 , . . . , ω d . Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting W to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

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Section: Introductionmentioning

confidence: 99%

Renormalization of vector fields and Diophantine invariant tori

Koch

Kocić

2008

Ergod. Th. Dynam. Sys.

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“…The n-th step RG transformation N n involves a change of variables (q, y) → (T n q, y), and another change of variables of the form (1.5), which eliminates certain "nonresonant modes". This is similar in spirit to the RG transformations used in [11][12][13][14][15][16][17][18][19]. The details of the elimination procedure can be found in Section 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 73%

Renormalization of diophantine skew flows, with applications to the reducibility problem

Koch¹,

Dias

2008

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex n×n matrices, for arbitrary Diophantine frequency vectors in R d and dimensions d,n. In cases where the group component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each Diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular 2×2 matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class C γ , with γ depending on the number of independent frequencies, and on the Diophantine exponent.

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“…A RG analysis of flows on T 2 with Brjuno rotation numbers was given in [96]. A RG analysis of one-parameter families of Hamiltonians with a pair of (golden) invariant tori bifurcating from a shearless invariant torus was carried out by Gaidashev in [54]. A brief description of this work, which involves two different fixed points of R, will be given in Subsection 5.5.…”

Section: Some Ideas and Resultsmentioning

confidence: 99%

“…Such tori appear in many physical systems, including models of the atmosphere, toroidal plasma devices, channel flows, and others [88,29,28,7,65,12]. Just like regular KAM tori, a shearless torus with Diophantine rotation number persists under small perturbations (of an integrable system) [32,54]. In fact, they are surprisingly stable.…”

Section: Non-twist Flowsmentioning

confidence: 99%

Renormalization of vector fields

Koch¹

2008

Holomorphic Dynamics and Renormalization

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Abstract. These notes cover some of the recent developments in the renormalization of quasiperiodic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on T d ×R . After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and invariant tori, including shearless tori, and non-differentiable tori for critical Hamiltonians. Then we discuss properties related to the spectrum of the linearized renormalization transformation, such as the accumulation rates for sequences of closed orbits. The last part describes extensions from "self-similar" to Diophantine rotation vectors. This involves sequences of renormalization transformations that are related to continued fractions expansions in one and more dimensions. Whenever appropriate, the discussion of details is restricted to special cases where inessential technical complications can be avoided.

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Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori

Cited by 18 publications

References 0 publications

Renormalization of vector fields and Diophantine invariant tori

Renormalization of vector fields and Diophantine invariant tori

Renormalization of diophantine skew flows, with applications to the reducibility problem

Renormalization of vector fields

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