V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84 (1984) (Russian). Revised English version: Mathematical Physics Preprint Archive 98-421,  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of the area preserving H enon map split transversally [9, 10, 4] The rest of the paper contains an informal derivation of the asymptotic formula (3) It can not be considered as a complete proof. The approach is based on Lazutkin s method originally proposed for a study of the standard map [13, 8]. Section 5 provides a summary of numerical experiments used to evaluate the splitting constant for some model families and to check the correctness of the asymptotic formula (3) 2 Formal interpolation by a ow In this section we discuss the formal interpolation of the family F by an ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84 (1984) (Russian). Revised English version: Mathematical Physics Preprint Archive 98-421,

....equations with slow dynamics are related with near the identity di eomorphisms by means of the Poincar e map. It turns out that being Hamiltonian is very important to get exponentially smallness. The Hamiltonian character of the equation is translated to the symplectic character of the maps. In [La1] Lazutkin studied the standard map F (x; y) x y sin x; y sin x) and provided the following formula for the angle between the stable and unstable manifolds at a homoclinic point = j 1 je [1 O( b ) 2) with 0 b 1=8. This was the rst exponentially small asymptotic formula ....

....followed basically the structure of [DS2] However, due to the fact that many of their arguments strongly rely on the hyperbolic character of the xed point, we have had to introduce new techniques to deal with the parabolic case. To this end we have also used tools introduced by Lazutkin [La2] [La1]. It is worth remarking that most of our arguments are can be adapted for the hyperbolic case. The memoir is organized as follows. In the rst chapter we introduce the notation, the hypotheses and the main theorem. In the second chapter, we study some analytical properties of the homoclinic orbit ....

V.F. Lazutkin, Splitting of separatrices for the Chirikov's standard map, VINITI 6372/84 (1984). (English version in www.maia.ub.es/mp arc #98-421.)

....separatrix (t) Then the stable separatrix can be represented in the parametric form, T ; E) t (t) t) 18 If it was entire, we could use arguments of Sect. 7 to show nonexistence of homoclinic connection. 19 The proof is based on the ideas rst formulated by V. F. Lazutkin [Laz84]. A rst general exponentially small upper bound for the splitting for close to identity maps was obtained by E. Fontich and C. Sim o [FS90b] in a closely related way. 30 where by de nition (t) T ( t) t ; t) E( t) 20) These function are periodic. Indeed, t ....

....point z 0 (the rst intersection of the separatrices with the line x = and the splitting angle is given by = 0 3 e 2 = 1 O( The lobe area is given by S = 2 0 e 2 = 1 O( The formula for the splitting angle was rst obtained by V. F. Lazutkin [Laz84]. The original proof was based on a detailed study of the analytic continuation of the function (T ) see the previous section) Lazutkin s proof was not complete. A complete and self contained proof of the formulas for the splitting of separatrices for the standard map can be found in [Gel99] ....

V. F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84, 1984. (Russian).

.... Theta;3 C Theta;2 ) C u;s 5 C u;s 6 ) 4:5:71) Taking into account the expressions for the Hamiltonians H 0 ; H 1 , one can show that C Theta;5 2 Delta 10 8 : 41 The following lemma provides an instrument for obtaining exponentially fine estimates on the real axis. Lemma 4.5. 2 ([11]) Assume ae 0 and b ae are given. If a 2ae Gammaperiodic function f is analytic and bounded by a constant in the complex strip jImzj b and has zero mean value, then on the real axis jf(t)j 2e e Gamma 1 sup jImzj b jf(z)j exp Gamma b ae ; jf 0 (t)j 2e 2 (e Gamma 1) ....

V.F.Lazutkin, Splitting of separatrices for the Chirikov's standard map, Preprint (1984), VINITI 6372/84.

....was widely used as a model for the motion near a resonance for Hamiltonian systems with two degrees of freedom [3] After a proper scaling in the vertical direction separatrices of the standard map look like the pendulum separatrix. Unlike the pendulum the separatrices of the standard map split [3,15], i.e. they intersect each other transversely. The splitting angle ff at a symmetrical primary homoclinic point is exponentially small: ff =4, 4j Theta 1 j h 2 e Gamma 2 =h O i e Gamma 2 =h j ; where h p is the logarithm of the multiplicator of the hyperbolic fixed ....

....(5) represent a formal separatrix. In particular, this shows that the splitting is less than any power of the small parameter h. In order to detect the splitting we study analytical continuation of the separatrices. In the case of the standard map this procedure was described in details in [15,9]. The series (5) also approximate the analytical continuation of the parametrizations for some complex values of t. When t approaches the singularities of the coefficients the error increases. The coefficients are 2i periodic and real analytic. Because of these symmetries it is sufficient to study ....

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V. F. Lazutkin, Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84 (1984), (Russian).

.... ( and then only two corresponding terms dominate in Fourier series. Theorem 2 For p 3 and small enough the invariant manifolds split, and the value of the splitting is predicted correctly by the Melnikov method. The method used for the proof is based on the ideas proposed by Lazutkin [4] for the study of the separatrix splitting for the standard map, adapted later to di erential equations [1, 2] We use a convergent Birkho normal form in a neighborhood of the hyperbolic torus. The normal form theorem is similar to Moser s theorem [5] on the normal form near a periodic hyperbolic ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI No. 6372-84 (in Russian), 1984.

....studies [DR99] showed that in some other simple degenerate cases Melnikov method also provides an accurate estimate for the splitting. The case n = 1 was considered previously in the paper [Gel00] In order to prove asymptotic formulae (1) 2) the author used the method proposed by Lazutkin [Laz84] and developed later by Lazutkin and his collaborators for the study of the separatrix splitting of the standard map (see [Gel99] where a detailed description of the method is given, and [Gel00] for a shorter exposition in context of an application to a bifurcation problem) The method is based ....

V. F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84, 1984, (Russian).

....of hyperbolic objects due to the breakdown of resonant invariant curves, the splitting of separatrices and the chaotic behaviour associated to it, the existence and breakdown of irrational invariant curves, etcetera. We can refer to the extense literature on the topic (see, for instance, 1] [8] and [11] If we take V j 0 in (1) we obtain an integrable twist map: q; p) 2 S 1 Theta R 7 (q p; p) 2 S 1 Theta R: 3) For any p 0 2 R, we have that S 1 Theta fp 0 g is an invariant circle of (3) and that the dynamics on the variable q is a rotation of angle j p 0 . If we ....

V.F. Lazutkin, "Splitting of Separatrices for the Chirikov's Standard Map", VINITI 6372/84. Preprint (1984).

....was widely used as a model for the motion near a resonance for hamiltonian systems with two degrees of freedom [3] After a proper scaling in the vertical direction separatrices of the standard map look like the pendulum separatrix. Unlike the pendulum the separatrices of the standard map split [3, 10], i.e. they intersect each other transversaly. The splitting angle, ff, at a symmetrical primary homoclinic point is exponentially small: ff =4, 4j Theta 1 j h 2 e Gamma 2 =h O i e Gamma 2 =h j ; where h p is the logarithm of the multiplicator of the hyperbolic fixed ....

....(5) represent a formal separatrix. In particular, this shows that the splitting is less than any power of the small parameter. In order to detect the splitting we study analytical continuation of the separatrices. In the case of the standard map this procedure was described with more details in [10, 6]. The series (5) also approximate the analytical continuation of the parametrizations on the complex values of t. When t approaches the singularities of the coefficients the error increases. The coefficients are 2i periodic and real analytic. Because of these symmetries it is sufficient to study ....

[Article contains additional citation context not shown here]

V. F. Lazutkin, Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84 (1984), (Russian).

....as well as the case of more than two perturbing frequencies, is more complicated. In the following sections we sketch the method used to justify that the prediction given by the Melnikov function is correct. The method used here is a generalization to the quasiperiodic case of the method used in [Laz84], DS92] Gel93] 3. Normal form and local manifolds The first step is to give a description of the dynamics near the 2D dimensional invariant torus T . So, we will show the existence of a convergent normal form in a neighbourhood of T . As we have seen during the analysis of the Melnikov ....

V. F. Lazutkin, Splitting of separatrices for the Chirikov's standard map, Preprint VINITI No. 6372--84 (in Russian), 1984.

....maps, exponentially small upper estimates of the splitting of separatrices have been obtained by several authors [Nei84, FS90, Fon95, Gel96] whereas the effective measure of the splitting size has only been formulated for some celebrated entire standard like maps by V. Lazutkin and co workers [Laz84, LST89, GLT91, GLS94], as well as by D. Treschev [Tre96a] However, a complete proof of the asymptotic formulae for the splitting of separatrices for these entire maps has not been published yet, in spite of the intensive efforts devoted to it. An important complexity arises from the fact that these celebrated maps ....

....also discussed numerically. To the best of our knowledge, this is the first time that such a formula is rigorously proved for a discrete dynamical system. The proof is based on a rigorous justification of the Melnikov method for maps. This methodology follows an approach suggested by V. Lazutkin [Laz84], which has been already developed for rapidly forced flows close to integrable ones [DS92, DS97] The model In the present paper we consider the family F : R 2 R 2 of standard like maps F (x; y) y; Gammax 2y 1 y 2 V 0 (y) cosh h; h 0; 2 R; 1.1) ....

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V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint 6372--84, VINITI, 1984. (In Russian).

.... [48] will not be dealt here, in spite of their crucial interest for the problem of the Arnold diffusion, for which we refer to the lecture by Pierre Lochak [31] The method we present here to handle the singular separatrix splitting for Hamiltonian flows was initiated by Lazutkin and co workers [29, 25], and it is based on the construction of a splitting function which is invariant under the action of the perturbed flow. The analyticity of the problem is pushed forward to compute this splitting function for complex values, and to recover it in the real world in form of an exponentially small in ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint 6372--84, VINITI, 1984. (In Russian).

....is convenient to formulate it as splitting size = e Gammafi=h Theta(h) Theta(h) bounded when h 0 : 1.1) Singular splitting and the Melnikov method 5 The next step was the attainment of exponentially small asymptotic formulae in some standard like maps, by V. Lazutkin and co workers [Laz84, LST89, GLT91], see also [HM93, Sur94, Tre96] For instance, regarding the standard map and the H enon map, in these works it is claimed that the splitting has an asymptotic behavior of the form 0 h fl exp( Gammafi=h) for some constants 0 6= 0, fi 0, and fl, that is, splitting size = h fl e ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI No. 6372--84 (in Russian), 1984.

....of detecting the splitting of separatrices from the Melnikov integrals is much more intricate because of its exponentially small character with respect to . Concerning the singular problem for 2 degrees of freedom (n = 1 in section 1. 5) or equivalently area preserving maps, the first results [43, 31] appeared in 1984, and now it seems fairly well understood (see, for instance, 22] for a survey of results) For more than 2 degrees of freedom, the existence of intersection between the whiskers can be detected by geometrical and topological methods [25, 8] but the effective measure of such ....

V. Lazutkin, Splitting of separatrices for the Chirikov's standard map (in Rus- HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS 27 sian). Preprint VINITI 6372-84, 1984.

....as well as the case of more than two perturbing frequencies, is more complicated. In the following sections we sketch the method used to justify that the prediction given by the Melnikov function is correct. The method used here is a generalization to the quasiperiodic case of the method used in [Laz84, DS92, Gel93]. 3. Normal Form and Local Manifolds The first step is to give a description of the dynamics near the 2D dimensional invariant torus T . So, we will show the existence of a convergent normal form in a neighbourhood of T . As we have seen during the analysis of the Melnikov function, the size of ....

V.F. Lazutkin, Splitting of separatrices for the Chirikov's standard map, Preprint VINITI No. 6372--84 (in Russian), 1984.

....families of maps, exponentially small upper estimates of the splitting of separatrices have been obtained by several authors [Nei84,FS90,Fon95,Gel96] whereas the effective measure of the splitting size has only been formulated for some celebrated standard like maps by V. Lazutkin and co workers [Laz84,LST89,GLT91,GLS94], as well as by D. Treschev [Tre96a] However, a complete proof of these asymptotic formulae has not been published yet, in spite of the intensive efforts devoted to it. Besides, all the maps for which there exist exponentially small asymptotic formulae for the splitting are of polynomial kind ....

....justification of the Melnikov method by studying the perturbed invariant curves of the maps (1.1) for complex values of the discrete time t, as close as possible to the singularities of the unperturbed natural parameterization z 0 (t) given in (1. 6) This approach was suggested by Lazutkin [Laz84] several years ago, for the case of the standard map. Our measure of the splitting of the separatrices is given by the so called splitting function Psi(s) defined in (2.22) as the graphic of the unstable curve over the stable one, in some flow box canonical coordinates. It is a h periodic ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI No. 6372--84 (in Russian), 1984.

....F ( and then only two corresponding terms dominate in Fourier series. Theorem 2 For p 3 and small enough the invariant manifolds split, and the value of the splitting is predicted correctly by the Melnikov method. The method used for the proof is based on the ideas proposed by Lazutkin [4] for the study of the separatrix splitting for the standard map, adapted later to differential equations [1, 2] We use a convergent Birkhoff normal form in a neighborhood of the hyperbolic torus. The normal form theorem is similar to Moser s theorem [5] on the normal form near a periodic ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI No. 6372--84 (in Russian), 1984.

....T T c is described by a trivial fixed point; a non trivial fixed point describes the case T = T c and another low 27Novembre1998 Draft #20 temperature fixed point describes the cases T T c . Evidence in this direction comes also from the theory of the standard map and its developments, [La,Gel2,Gel1]. Acknowledgments. This work is part of the research program of the European Network on Stability and Universality in Classical Mechanics , # ERBCHRXCT940460. Supported in part by CNR GNFM and Rutgers University. Appendix A1. Counterterms A1.1 To order h one can write, by using (2.12) 2.25) ....

Lazutkin, V.F.,, Splitting of separatrices for the Chirikov's standard map, mp arc@ math.utexas.edu, #98-- 421, (annotated english translation of a 1984 paper).

....the Melnikov function and the splitting of separatrices, for a lower value of the exponent p. The rest of the paper is devoted to the proof of the Main Theorem. In contrast with the above mentioned papers, the method used in the present paper is based on the geometrical ideas proposed by Lazutkin [Laz84] for the study of the separatrix splitting for the standard map, and adapted to differential equations by the authors [Gel90, DS92, Gel93] In section 2, the Melnikov function is carefully analyzed, to provide its asymptotic behaviour. In section 3, like in [DS92, DS96] and as a first step to ....

V.F. Lazutkin. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI No. 6372--84 (in Russian), 1984.

....OF SEPARATRICES FOR THE CHIRIKOV S STANDARD MAP V.F.Lazutkin Abstract. This is a revised English version of my earlier paper [L1] deposited in VINITI in 1984. An asymptotic formula for the exponentially small angle of intersection of the stable and unstable manifolds of the fixed hyperbolic point for the standard map is derived provided the parameter tends to zero. x1. Introduction. The formula for the separatrices ....

....small angle of intersection of the stable and unstable manifolds of the fixed hyperbolic point for the standard map is derived provided the parameter tends to zero. x1. Introduction. The formula for the separatrices splitting This paper is a revised English version of my earlier paper [L1] deposited in VINITI in 1984. Here I changed some notations, changed the formulations of the conjectures to make them more realistic (the first one now is a theorem, the second one remains open) and added recent references. I keep the numbers for old references while the new ones acquired letter ....

[Article contains additional citation context not shown here]

]V. F. LAZUTKIN, Splitting of separatrices for the Chirikov's standard map., Preprint (1984), VINITI 6372/84.

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