scholarly journals Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Nonlinearity ◽  
2015 ◽  
Vol 28 (7) ◽  
pp. 2555-2585 ◽  
Author(s):  
Guido Gentile
Keyword(s):  
Invariant Curves ◽  
Rotation Numbers ◽  
Twist Maps

scholarly journals Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

1988 ◽  
Vol 8 (4) ◽  
pp. 555-584 ◽  
Author(s):  
Raphaël Douady
Keyword(s):  
Hamiltonian Systems ◽  
Rotation Number ◽  
Positive Measure ◽  
Invariant Curves ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractWe prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

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

scholarly journals The non-hyperbolicity of irrational invariant curves for twist maps and all that follows

2016 ◽  
Vol 32 (4) ◽  
pp. 1295-1310 ◽  
Author(s):  
Marie-Claude Arnaud ◽  
Pierre Berger
Keyword(s):  
Invariant Curves ◽  
Twist Maps

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 Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

Propriétés des attracteurs de Birkhoff

1988 ◽  
Vol 8 (2) ◽  
pp. 241-310 ◽  
Author(s):  
P. Le Calvez
Keyword(s):  
Periodic Point ◽  
Infinite Number ◽  
Twist Map ◽  
Rotation Numbers ◽  
Twist Maps

AbstractWe study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

scholarly journals Continuity of the Peierls barrier and robustness of laminations

2014 ◽  
Vol 35 (4) ◽  
pp. 1263-1288 ◽  
Author(s):  
BLAŽ MRAMOR ◽  
BOB RINK
Keyword(s):  
Broad Class ◽  
Variational Problems ◽  
Similar Estimate ◽  
Solid State Physics ◽  
Research Workshop ◽  
Rotation Numbers ◽  
Peierls Barrier ◽  
Twist Maps ◽  
The Difference ◽  
Math Phys

AbstractWe study the Peierls barrier$P_{\omega }(\xi )$for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start by deriving an estimate for the difference$\vert P_{\omega }(\xi ) - P_{q/p}(\xi ) \vert $of the Peierls barriers of rotation numbers$\omega \in {{\mathbb{R}}}$and$q/p\in {\mathbb{Q}}$. A similar estimate was obtained by Mather [Modulus of continuity for Peierls’s barrier.Proc. NATO Advanced Research Workshop on Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, Italy, 13–18 October 1986) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 209).Eds. P. H. Rabinowitz, A. Ambrosetti and I. Eckeland. D. Reidel, Dordrecht, 1987, pp. 177–202] in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that$\omega \mapsto P_{\omega }(\xi )$is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers of rotation number$\omega \in {{\mathbb{R}}}\delimiter "026E30F {\mathbb{Q}}$is open in the$C^1$-topology.

A method for proving that monotone twist maps have no invariant circles

1991 ◽  
Vol 11 (1) ◽  
pp. 79-84 ◽  
Author(s):  
Irwin Jungreis
Keyword(s):  
Existence Theorem ◽  
Computer Assisted ◽  
Twist Map ◽  
Standard Map ◽  
Rotation Numbers ◽  
Invariant Circles ◽  
Twist Maps ◽  
Parameter Values

AbstractWe present an existence theorem for certain kinds of orbits of a monotone twist map and use it to obtain a criterion for proving that there are no invariant circles with a certain range of rotation numbers. We have used this criterion to prove (computer assisted) that the standard map has no invariant circles for several parameter values includingk= 0.9718.

Diffeomorphisms with infinitely many irrational invariant curves

2010 ◽  
Vol 31 (5) ◽  
pp. 1517-1535
Author(s):  
LEONARDO MORA ◽  
BLADISMIR RUIZ
Keyword(s):  
Saddle Points ◽  
Irrational Rotation ◽  
The Other ◽  
Heteroclinic Cycle ◽  
Invariant Curves ◽  
Open Set ◽  
Rotation Numbers ◽  
Surface Diffeomorphism ◽  
Dense Set

AbstractFor a surface diffeomorphism f∈Diff l(M), with l≥8, we prove that if f exhibits a non-transversal heteroclinic cycle composed of two fixed saddle points Q1 and Q2, one dissipative and the other expansive, then there exists an open set 𝒱⊂Diff l(M) such that $ f \in \overline {\mathcal {V}}$ and there exists a dense set 𝒟⊂𝒱 such that for all g∈𝒟, g exhibits infinitely many invariant periodic curves with irrational rotation numbers. Moreover, these curves are C1 conjugated to an irrational rotation on 𝕊1.

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