Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models

1992 ◽  
Vol 56 (2-3) ◽  
pp. 123-134 ◽  
Author(s):  
S. Aubry ◽  
R.S. MacKay ◽  
C. Baesens
Keyword(s):  
Twist Maps ◽  
Uniform Hyperbolicity

scholarly journals Kupka–Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

Nonlinearity ◽  
2017 ◽  
Vol 30 (10) ◽  
pp. 3895-3931
Author(s):  
Renaud Leplaideur ◽  
Isabel Lugão Rios
Keyword(s):  
Uniform Hyperbolicity

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

HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

2001 ◽  
Vol 11 (09) ◽  
pp. 2451-2461
Author(s):  
TIFEI QIAN
Keyword(s):  
Variational Principle ◽  
Variational Method ◽  
Fixed Points ◽  
Heteroclinic Orbit ◽  
Geometric Method ◽  
Heteroclinic Orbits ◽  
Constructive Approach ◽  
Connecting Orbits ◽  
Relative Ease ◽  
Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

scholarly journals Birkhoff periodic orbits for twist maps with the graph intersection property

1985 ◽  
Vol 5 (4) ◽  
pp. 531-537 ◽  
Author(s):  
David Bernstein
Keyword(s):  
Periodic Orbits ◽  
Intersection Property ◽  
Twist Maps

AbstractIn this paper we show that Birkhoff periodic orbits actually exist for arbitrary monotone twist maps satisfying the graph intersection property.

scholarly journals Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

2014 ◽  
Vol 36 (1) ◽  
pp. 215-255 ◽  
Author(s):  
SAMUEL SENTI ◽  
HIROKI TAKAHASI
Keyword(s):  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Statistical Properties ◽  
Limit Set ◽  
Bifurcation Parameter ◽  
Large Interval ◽  
Equilibrium Measures ◽  
Uniform Hyperbolicity ◽  
Unstable Direction

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential$-t\log J^{u}$, where$t\in \mathbb{R}$is in a certain large interval and$J^{u}$denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

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

Complete orbits for twist maps on the plane: the case of small twist

2010 ◽  
Vol 31 (5) ◽  
pp. 1471-1498 ◽  
Author(s):  
MARKUS KUNZE ◽  
RAFAEL ORTEGA
Keyword(s):  
Moving Wall ◽  
Ping Pong ◽  
Twist Maps ◽  
Unbounded Orbits

AbstractIn this article we consider twist maps that are non-periodic (and hence are defined on the plane rather than on the cylinder) and have small twist at infinity. Under natural assumptions the existence of infinitely many bounded orbits is established, and furthermore it is proved that unbounded orbits follow bounded orbits for long times. An application is given to the Fermi–Ulam ping-pong model with a non-periodic moving wall.

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.

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