scholarly journals Invariant circles and rotation bands in monotone twist maps

1987 ◽  
Vol 113 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Philip L. Boyland
Keyword(s):  
Invariant Circles ◽  
Twist Maps

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.

Analytic destruction of invariant circles

1994 ◽  
Vol 14 (2) ◽  
pp. 267-298 ◽  
Author(s):  
Giovanni Forni
Keyword(s):  
Small Perturbations ◽  
Invariant Circles ◽  
Twist Maps ◽  
Area Preserving

AbstractWe give destruction results under analytic small perturbations for invariant circles of exact area-preserving monotone twist maps, applying methods developed by M. Herman and J. Mather.

scholarly journals Invariant circles and the order structure of periodic orbits in monotone twist maps

Topology ◽  
1987 ◽  
Vol 26 (1) ◽  
pp. 21-35 ◽  
Author(s):  
Philip L. Boyland ◽  
Glen Richard Hall
Keyword(s):  
Periodic Orbits ◽  
Order Structure ◽  
Invariant Circles ◽  
Twist Maps

scholarly journals About periodic and quasi-periodic orbits of a new type for twist maps of the torus

2002 ◽  
Vol 74 (1) ◽  
pp. 25-31
Author(s):  
SALVADOR ADDAS-ZANATA
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Classical Result ◽  
Important Class ◽  
Twist Map ◽  
Positive Topological Entropy ◽  
Invariant Circles ◽  
Twist Maps ◽  
New Type ◽  
Rotational Invariant

We prove that for a large and important class of C¹ twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (R.I.C's). These orbits have a non-zero "vertical rotation number'' (V.R.N.), in contrast to what happens to Birkhoff periodic orbits and Aubry-Mather sets. The V.R.N. is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a V.R.N = a > 0, implies the existence of orbits with V.R.N = b, for all 0 < b < a. And as a consequence of the previous results we get that a twist map of the torus with no R.I.C's has positive topological entropy, which is a very classical result. In the end of the paper we present some applications and examples, like the Standard map, such that our results apply.

scholarly journals Quantitative destruction of invariant circles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lin Wang
Keyword(s):  
Trigonometric Polynomial ◽  
Integrable System ◽  
Arithmetic Property ◽  
Invariant Circles ◽  
Twist Maps ◽  
Area Preserving

<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id="M3">\begin{document}$ R_N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \|R_N\|_{C^r}&lt;\epsilon $\end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id="M8">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id="M9">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Non-twist invariant circles in conformally symplectic systems

2021 ◽  
Vol 96 ◽  
pp. 105695
Author(s):  
Renato Calleja ◽  
Marta Canadell ◽  
Alex Haro
Keyword(s):  
Invariant Circles

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.

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