Birkhoff-Hénon attractors for dissipative perturbations of area-preserving twist maps

1994 ◽  
Vol 14 (4) ◽  
pp. 807-815 ◽  
Author(s):  
Leonardo Mora
Keyword(s):  
Recent Result ◽  
Invariant Curve ◽  
Twist Map ◽  
Saddle Node ◽  
Twist Maps ◽  
Area Preserving

AbstractWe prove that an area-preserving twist map having an invariant curve, can be approximated by a twist map exhibiting a Birkhoff-Hénon attractor. This is done by showing that the invariant curve can be perturbed into a saddle-node cycle with criticalities and by using a recent result reported by Diaz, Rocha and Viana.

scholarly journals A Topological Approach to Bend-Twist Maps with Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Anna Pascoletti ◽  
Fabio Zanolin
Keyword(s):  
Fixed Point Theorems ◽  
Small Perturbations ◽  
Topological Approach ◽  
Twist Map ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Planar Systems ◽  
Twist Theorem ◽  
Area Preserving

In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

scholarly journals Boundaries of instability zones for symplectic twist maps

2013 ◽  
Vol 13 (1) ◽  
pp. 19-41 ◽  
Author(s):  
M.-C. Arnaud
Keyword(s):  
Rotation Number ◽  
Irrational Rotation ◽  
Invariant Curve ◽  
Partial Answer ◽  
Twist Map ◽  
Instability Zone ◽  
Counter Example ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
The One

AbstractVery few things are known about the curves that are at the boundary of the instability zones of symplectic twist maps. It is known that in general they have an irrational rotation number and that they cannot be KAM curves. We address the following questions. Can they be very smooth? Can they be non-${C}^{1} $?Can they have a Diophantine or a Liouville rotation number? We give a partial answer for${C}^{1} $and${C}^{2} $twist maps.In Theorem 1, we construct a${C}^{2} $symplectic twist map$f$of the annulus that has an essential invariant curve$\Gamma $such that$\bullet $ $\Gamma $is not differentiable;$\bullet $the dynamics of${f}_{\vert \Gamma } $is conjugated to the one of a Denjoy counter-example;$\bullet $ $\Gamma $is at the boundary of an instability zone for$f$.Using the Hayashi connecting lemma, we prove in Theroem 2 that any symplectic twist map restricted to an essential invariant curve can be embedded as the dynamics along a boundary of an instability zone for some${C}^{1} $symplectic twist map.

scholarly journals Some remarks on Birkhoff and Mather twist map theorems

1982 ◽  
Vol 2 (2) ◽  
pp. 185-194 ◽  
Author(s):  
A. Katok
Keyword(s):  
Recent Result ◽  
Periodic Orbits ◽  
Invariant Sets ◽  
Classical Theorem ◽  
Twist Map ◽  
Twist Maps ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractA recent result of J. Mather [1] about the existence of quasi-periodic orbits for twist maps is derived from an appropriately modified version of G. D. Birkhoff's classical theorem concerning periodic orbits. A proof of Birkhoff's theorem is given using a simplified geometric version of Mather's arguments. Additional properties of Mather's invariant sets are discussed.

scholarly journals Hyperbolic sets for twist maps

1985 ◽  
Vol 5 (3) ◽  
pp. 337-339 ◽  
Author(s):  
Daniel L. Goroff
Keyword(s):  
Hyperbolic Structure ◽  
Twist Map ◽  
Twist Maps ◽  
Hyperbolic Sets ◽  
Area Preserving

AbstractAn example is given of an area-preserving monotone twist map such that a uniformly hyperbolic structure exists on the closure of its Birkhoff maximizing orbits.

scholarly journals A topological version of a theorem of Mather on twist maps

1984 ◽  
Vol 4 (4) ◽  
pp. 585-603 ◽  
Author(s):  
Glen Richard Hall
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Point ◽  
Rotation Number ◽  
Point Theorem ◽  
Twist Map ◽  
Twist Maps ◽  
Recent Theorem ◽  
Topological Version ◽  
Area Preserving

AbstractIn this report we show that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. We use topological techniques so no assumption of area-preservation or circle intersection property is needed. If the map is area-preserving then this theorem andthe fixed point theorem of Birkhoff imply a recent theorem of Aubry and Mather. We also show that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff.

Invariant curves for area-preserving twist maps far from integrable

1991 ◽  
Vol 65 (3-4) ◽  
pp. 617-643 ◽  
Author(s):  
Alessandra Celletti ◽  
Luigi Chierchia
Keyword(s):  
Invariant Curves ◽  
Twist Maps ◽  
Area Preserving

Rigidity of integrable twist maps and a theorem of Moser

1998 ◽  
Vol 18 (3) ◽  
pp. 725-730
Author(s):  
KARL FRIEDRICH SIBURG
Keyword(s):  
Higher Dimensions ◽  
Point Of View ◽  
Compact Set ◽  
Rigidity Result ◽  
Twist Map ◽  
Twist Maps

According to a theorem of Moser, every monotone twist map $\varphi$ on the cylinder ${\Bbb S}^1\times {\Bbb R}$, which is integrable outside a compact set, is the time-1-map $\varphi_H^1$ of a fibrewise convex Hamiltonian $H$. In this paper we prove that if this particular flow $\varphi_H^t$ is also integrable outside a compact set, then $\varphi$ has to be integrable on the whole cylinder (and vice versa, of course). From this dynamical point of view, integrable twist maps appear to be quite rigid.As is shown in the appendix, an analogous rigidity result becomes trivial in higher dimensions.

Construction of invariant measures supported within the gaps of Aubry–Mather sets

1996 ◽  
Vol 16 (1) ◽  
pp. 51-86 ◽  
Author(s):  
Giovanni Forni
Keyword(s):  
Variational Approach ◽  
Invariant Measures ◽  
Positive Entropy ◽  
Almost Periodic ◽  
Invariant Curves ◽  
Mather Theory ◽  
Entropy Measures ◽  
Twist Maps ◽  
Minimization Procedure ◽  
Area Preserving

AbstractThis paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.

scholarly journals On the Measure-Preserving Mappings with Three-Dimensions

1993 ◽  
Vol 132 ◽  
pp. 73-89
Author(s):  
Yi-Sui Sun
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Invariant Curve ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Kolmogorov Entropy ◽  
Numerical Exploration ◽  
Area Preserving ◽  
Dimensional Mapping

AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

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