Serge J. Aubry. The concept of anti-integrability: de nition, theorems and applications to the standard map. In Twist mappings and their applications, volume 44 of IMA Vol. Math. Appl., pages 7-54. Springer, New York, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....based on variational methods, is still not published. An unusual set up (quite far from the original Arnold s one) was proposed by Easton, Meiss and Roberts [16] They consider di usion in a system, which is not near integrable. On the contrary, the system is close to the anti integrable limit [2, 3, 20]. Because of a strong hyperbolic properties of the system the construction of di usion trajectories turns out to be simple and elegant. In this paper we deal with the so called a priori unstable systems. The terminology is taken from [9] where the systems (1.1) are called a priori stable. As ....

Aubry S., The concept of anti-integrability: De nition, theorems and applications to the standard map. Twist mappings and their applications, Ed R.McGehee, K.R.Meyer, P. 7-54, 1992.

....that the action A( A(ae) ffl. We then argue why we believe that the condition of the finiteness of the distance function in a small neighbourhood of M ae is typically (possibly always) satisfied. In particular, we show that it is satisfied in the anti integrable limit in the sense of Aubry [Aub92]. The paper is structured as follows: In section 2 we discuss three standing assumptions on the generating function h of a given Lagrangian map: A1) periodicity, A2) the superlinear growth, and (A3) the existence of the semiflow OE . We show that (A3) is satisfied if e.g. the second ....

....2 S E , such that ae( ae, and A( fi(ae) ffl. 7.4 Anti integrable limit with non degenerate potential. In this section we construct a family of examples which satisfy the conditions of Theorem 7.6. The examples are symplectic maps very far from integrable; in the sense called by Aubry [Aub92] the anti integrable limit. Definition 7.1 We say that a C 1 function P : R d R satisfying P (x a) P (x) for every a 2 Z d is a non degenerate potential, if there exist constants D; ffl 0 such that for every C 1 curve fl : 0; T ] R d satisfying for all t 2 (0; T ) inf j 0 ....

S. Aubry. The concept of anti-integrability: definition, theorems and applications to the standard map. in Twist mappings and their applications. Springer, New York, 1992.

.... standard map is x 0 = x 1 y 0 ; y 0 = y DV (x) 4) The case when the con guration space is one dimensional, n = 1, and V (x) k cos(x) corresponds to the standard map [Mei92] 3 Anti Integrable Limit A discrete dynamical system is said to have an anti integrable limit [Aub92, MM92, Aub95], when the dynamics reduces to a full shift on a discrete phase space. For example, the variational principle for the natural system (3) reduces to X t V (x t ) when = 0. In this case the Euler Lagrange equations reduce to DV (x t ) 0, which implies that any sequence of critical points ....

S. J. Aubry. The concept of anti-integrability: Denition, theorems and applications to the standard map. Twist Mappings and their Applications, Ed Richard McGehee, Kenneth R. Meyer, pages 7-54, 1992.

....dynamics weakly mixing but not mixing on a set of positive measure While Pesin theory together with a genericity result for shift invariant measures [77] provide many invariant measures which have this property, they are in general supported on sets of zero Lebesgue measure. This observation by [6] can be strengthened by estimating the Hausdorff dimension of the measure. It was first observed in [18] that there are invariant measures whose Hausdorff dimension goes to 2 for 1. This observation is based on Young s formula [153] for the Hausdorff dimension H( of : H( 2h (T ) ....

S. Aubry. The concept of anti-integrability: definition, theorems and applications to the Standard map. In K.Meyer R.Mc Gehee, editor, Twist mappings and their Applications, IMA Volumes in Mathematics, Vol. 44. Springer Verlag, 1992.

.... AMS classification scheme numbers: 58F05, 58F03, 58C15 1 Introduction The problem of determining the sequence of bifurcations that result in the creation of a Smale horseshoe is an interesting one [1, 2, 3] In this paper we use a continuation technique based on an anti integrable (AI) limit [4] to study some of these bifurcations from the opposite side, that is, as bifurcations that destroy the horseshoe. As a simple example, we study the two parameter H enon family of maps x 0 y 0 = y Gamma k x 2 Gammabx : 1) As we recall in x2, the AI limit for this map is ....

....not deterministic. In this case we say that ffl = 0 corresponds to an antiintegrable limit (AI) of the map f [18] If the derivative of H is nonsingular, then a straightforward implicit function argument can be used to show that the AI orbits can be continued for ffl 6= 0 to orbits of the map f [4, 19]. An AI limit with this property is called nondegenerate. For example, consider the H enon map Eq. 1) Denoting points on an orbit by a sequence x t ; t 2 Z, we can rewrite Eq. 1) as a second order difference equation x t 1 bx t Gamma1 k Gamma x 2 t = 0: Introducing the scaled ....

S. J. Aubry. The concept of anti-integrability: Definition, theorems and applications to the standard map. Twist Mappings and their Applications, Ed Richard McGehee, Kenneth R. Meyer, pages 7--54, 1992.

.... AMS classification scheme numbers: 58F05, 58F03, 58C15 1 Introduction The problem of determining the sequence of bifurcations that result in the creation of a Smale horseshoe is an interesting one [1, 2, 3] In this paper we use a continuation technique based on an anti integrable (AI) limit [4] to study some of these bifurcations from the opposite side, that is, as bifurcations that destroy the horseshoe. As a simple example, we study the family of H enon maps [5, 6] x 0 y 0 = y Gamma k x 2 Gammabx : 1) Apart from the fact that the H enon maps are the simplest, ....

....not deterministic. In this case we say that ffl = 0 corresponds to an antiintegrable (AI) limit of the map f [25] If the derivative of H is nonsingular, then a straightforward implicit function argument can be used to show that the AI orbits can be continued for ffl 6= 0 to orbits of the map f [4, 26]. An AI limit with this property is called nondegenerate. For example, consider the H enon map Eq. 1) Denoting points on an orbit by a sequence x t , t 2 Z, we can rewrite Eq. 1) as a second order difference equation x t 1 bx t Gamma1 k Gamma x 2 t = 0 : Introducing the scaled ....

S. J. Aubry. The concept of anti-integrability: Definition, theorems and applications to the standard map. Twist Mappings and their Applications, Ed Richard McGehee, Kenneth R. Meyer, pages 7--54, 1992.

....on the real line having as the support the spectrum oe(L(x) for almost all x 2 X . A critical point q is called hyperbolic, if its Hessian L is invertible. This is equivalent with 0 = 2 oe(L(x) for almost all x 2 X . The embedded system of a hyperbolic critical point is a hyperbolic set (see [Aub 92a] Therefore, we call an invariant measure defined by a hyperbolic critical point also hyperbolic and say also the embedding is hyperbolic, if is hyperbolic. If is a hyperbolic invariant measure, then it defines by the implicit function theorem a family of measures S 0 7 S 0 , for S 0 ....

....shift and the isomorphy classes of Bernoulli shifts is determined by the metric entropy, we have embedded the Bernoulli shift and not only a factor. 2 Remarks. 1) The idea to go to the limit jflj 1 is due to Aubry [Aub 90] and is called anti integrable limit. The idea has been used further in [Aub 92a, Aub 92b] 2) If q 0 (X) mod (2 Z) 2 and the metric entropy of (X; T ; m) is small enough, then E S fl (T ) is not empty. 3) Given k 2 [0; 1] and N 2 H(T ) If the dynamical system (X; T ; m) is aperiodic, there exists an immersion with Morse index k. We only have to choose the function q 0 such ....

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S.Aubry. The concept of anti-integrability: definition, theorems and applications to the standard map. In "Twist mappings and their Applications", IMA Volumes in Mathematics, Vol. 44., Eds. R.Mc Gehee, K.Meyer, 1992

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Serge J. Aubry. The concept of anti-integrability: de nition, theorems and applications to the standard map. In Twist mappings and their applications, volume 44 of IMA Vol. Math. Appl., pages 7-54. Springer, New York, 1992.

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