H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Ergodic Theory Dynam. Systems, 19:1-47, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....there is convergence to a limit set with a Gauss map dynamics on it. This is valid for diophantine frequency vectors. 1 Introduction A renormalisation operator R is defined for analytic vector fields on the torus of dimension two, associated with a frequency vector. This approach is based in [10] for Hamiltonian functions, and in [6] for flows on T =Z , d 2. The vector fields considered here are of Poincar e type, i.e. there is a classification by a unique winding ratio on which R acts as the Gauss map. The slope ff of the winding ratio is mapped by the shift of its continued ....

....winding ratio, attracts all the nearby orbits in the same homotopy class. This can be applied to other flows on domains with an extra vertical Email: j.lopes dias damtp.cam.ac.uk dimension. Also, it might be used in KAM theory for two degrees of freedom Hamiltonian systems, in order to extend [10]. The allowed set of frequencies is smaller that the one obtained by the usual KAM Theorem. Nevertheless, this is not a real disadvantage of our method since it is of equal full Lebesgue measure. Our procedure can be related to the one followed by MacKay [13] and Chandre and Moussa [4] for an ....

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H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Erg. Theor. Dyn. Syst., 19:475--521, 1999.

....asymptotically contracts locally any X into a (d Gamma 1) parameter family of constant vector fields that includes . The renormalisation consists of linear and non linear coordinate changes of T , and a time rescaling. The existence of the linear part, based on a number theory result used in [6], depends on . In particular, for d = 2, the class of such vectors corresponds to the set of vectors with a quadratic irrational slope. Specifically for the two dimensional case, it is defined in E mail: j.lopes dias damtp.cam.ac.uk [4] a family of renormalisation iterative schemes allowing ....

....all the perturbation terms because the linear transformation shifts resonant to non resonant terms. Eventually all perturbation terms are eliminated while iterating R. This idea of renormalising vector fields is due to MacKay [8] and we follow an approach partly inspired by the work of Koch [6] on renormalisation for d degrees of freedom analytic Hamiltonian systems. The latter apply to the problem of stability of invariant tori associated with a frequency vector . This paper is organised as follows. In Section 2 we recall the basic ingredients of equivalence of flows, and in Section ....

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H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Erg. Theor. Dyn. Syst., 19:475--521, 1999.

....universal scaling behavior of critical invariant tori. Keywords: renormalization group, breakup of invariant tori, KAM theory. 1 Introduction Recently, renormalization group ideas have been proposed to describe the breakup of invariant tori for Hamiltonian systems with two degrees of freedom [10, 8, 3, 4, 1]. Most of the numerical work has been done for the golden mean torus (mainly for practical convenience) In this article, we extend the renormalization group transformation to more general frequencies. We consider the following class of Hamiltonians with two degrees of freedom, quadratic in the ....

.... Delta Delta USn ffi Delta Delta Delta ; where I Gamma f = I Gamma g = 0. We notice that step (4) does not change 0 and Omega [as opposed to steps (1) 2) and (3) The transformation UH is rigorously defined for a sufficiently small perturbation, consisting of f and g (see Ref. [10]) but the convergence in the whole domain of existence of the torus is a conjecture based on numerical observations. In summary the renormalization transformation acts as follows: First, some of the resonant modes are turned into nonresonant ones (by a rescaling of phase space) Then a KAM type ....

H. Koch, A Renormalization Group for Hamiltonians, with Applications to KAM tori, Erg. Theory Dyn. Syst., to appear (1999).

....H H ffi T Gamma ffl ; 1:2) where , and ffl are allowed to depend on H. In what follows, ffl is taken to be zero. U some fixed canonical transformation, introduced for convenience later on. The canonical transformation 00 will be described below. It is similar to the one introduced in [11], but differs significantly from those used in earlier renormalization schemes [1 6] The problem that needs to be dealt with is that the map H 7 H involves a loss of regularity in the direction ( 0) due to the fact that T expands this direction. All other directions are contracted, if # ....

....find fixed points, and to study the action of R near these fixed points. The Hamiltonian H 0 is such a fixed point, with UH = I . The corresponding RG analysis (for Hamiltonians that are not necessarily even in q or satisfy 3 H = 1, but are analytic) yields results about invariant tori [11] and sequences of closed orbits accumulating at these tori [12] There are other integrable fixed points, such as H (q; p) Delta p ( Omega , with scaling = # 2 =# 1 . But more interestingly, numerical results [1 10] suggest that there exist non integrable fixed points as well, for d ....

H. Koch, A Renormalization Group for Hamiltonians, with Applications to KAM Tori. Erg. Theor. Dyn. Syst. 19, 1--47 (1999).

....conjugate to the linear flow q j (t) q j (0) tc j (mod 2) for some nonzero constant c. We are particularly interested in rotation vectors = 1; 2 ; d ) whose components span an algebraic number field of degree d. They can also be characterized by a self similarity property [11]: There exists an integer d Theta d matrix T , with determinant Sigma1 and d Gamma 1 simple eigenvalues of modulus less than 1, for which is an eigenvector with a real eigenvalue # 1 1. The best known example is = 1; # 1 ) with # 1 the golden mean, and T = Gamma 0 1 1 1 Delta . We ....

.... Gamma H = 0. It is straightforward to verify the following. Proposition 2. If ae Gamma ae 0 , oe and are positive and sufficiently small, then H 7 H ffi T defines a compact linear map from I A ae 0 to A ae . In fact, the sufficiently small can be replaced by explicit inequalities [11,12]. Motivated by this result, we try to define the canonical transformation UH in equation (2) in such a way that for every H in some suitable subset of A ae , 6) H ffi UH 2 A ae 0 ; I Gamma Gamma H ffi UH Delta = 0 : This is compatible with choosing UH = I whenever H(q; p) depends on p ....

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H. Koch, A renormalization group for Hamiltonians, with applications to KAM tori, Erg. Theor. Dyn. Syst. 19 (1999), 1--47.

....to the linear flow q j (t) q j (0) tc# j (mod 2#) for some nonzero constant c. We are particularly interested in rotation vectors # = 1, # 2 , # d ) whose components span an algebraic number field of degree d. They can also be characterized by a self similarity property [11]: There exists an integer d d matrix T , with determinant 1 and d 1 simple eigenvalues of modulus less than 1, for which # is an eigenvector with a real eigenvalue # 1 1. The best known example is # = 1, # 1 ) with # 1 the golden mean, and T = # 0 1 1 1 # . We note that the matrix ....

....if I H = 0. It is straightforward to verify the following. Proposition 2. If # # # , # and # are positive and su#ciently small, then H ## H # T defines a compact linear map from I A # # to A # . In fact, the su#ciently small can be replaced by explicit inequalities [11,12]. Motivated by this result, we try to define the canonical transformation UH in equation (2) in such a way that for every H in some suitable subset of A # , 6) H # UH # A # # , I # H # UH # = 0 . This is compatible with choosing UH = I whenever H(q, p) depends on p only, since by ....

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H. Koch, A renormalization group for Hamiltonians, with applications to KAM tori, Erg. Theor. Dyn. Syst. 19 (1999), 1--47.

....to the stability problem for invariant tori associated with frequency vectors = 1 ; 2 ) for which 1 = 2 is a reduced quadratic irrational. For simplicity, we will focus from the beginning on the case of the golden mean. Our formalism can also be extended to higher dimensions, as in [4], but the results may be qualitatively different; see also [15 17] We start with some simple facts about the golden mean # = 1 2 1 2 p 5. The numbers # and Gamma1=# are solutions of a quadratic equation with integer coefficients, namely the equation for the eigenvalues of the matrix T ....

....critical scaling. The fixed point equation R(H) H should of course imply self similarity for the set of # k orbits of H. A transformation which has the potential of satisfying all of the necessary requirements (modulo the existence of a trivial unstable direction) has been described in [4]. It is of the following form: H) H ffi UH ffi T Gamma ffl ; T (q; p) Gamma T q; T Gamma1 p Delta : 1:2) UH is a canonical change of coordinates that will be defined later; in particular, UH is the identity map if H(q; p) does not depend on q. The constants ffl = ffl(H) ....

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H. Koch, A Renormalization Group for Hamiltonians, with Applications to KAM Tori. Preprint U. Texas, mp arc 96--383 (1996), to appear in Erg. Theor. Dyn. Syst.

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H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Ergodic Theory Dynam. Systems, 19:1-47, 1999.

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H. Koch, A renormalization group for Hamiltonians, with applications to KAM tori, Erg. Theor. Dyn. Syst. 19 (1999), 475-521.

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H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Ergodic Theory Dynam. Systems, 19(2):475-521, 1999.

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H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Ergodic Theory Dynam. Systems, 19(2):475-521, 1999.

No context found.

H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori. Ergodic Theory Dynam. Systems, 19(2):475-521, 1999.

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