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.

ELLIPTIC NON-BIRKHOFF PERIODIC ORBITS IN THE TWIST MAPS

1993 ◽  
Vol 03 (01) ◽  
pp. 165-185 ◽  
Author(s):  
ARTURO OLVERA ◽  
CARLES SIMÓ
Keyword(s):  
Periodic Orbits ◽  
Periodic Points ◽  
Irrational Rotation ◽  
Point Of View ◽  
Twist Map ◽  
Qualitative And Quantitative ◽  
Open Set ◽  
Twist Maps ◽  
Rotational Curve ◽  
Irrational Rotation Number

We consider a perturbed twist map when the perturbation is big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant Cantorian set appears. From another point of view, the destruction of the IRC is associated with the appearance of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated with the existence of non-Birkhoff orbits. In this paper we relate the different approaches. In order to explain the creation of non-Birkhoff orbits, we provide qualitative and quantitative models. We show the existence of elliptic non-Birkhoff periodic orbits for an open set of values of the perturbative parameter. The bifurcations giving rise to the elliptic non-Birkhoff orbits and other related bifurcations are analysed. In the last section, we show a celestial mechanics example displaying the described behavior.

scholarly journals Twist sets for maps of the circle

1984 ◽  
Vol 4 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Michał Misiurewicz
Keyword(s):  
Rotation Number ◽  
Closed Set ◽  
Twist Map ◽  
Rotation Interval ◽  
Continuous Map ◽  
The One

AbstractLet f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.

A Necessary Condition of Nonminimal Aubry–Mather Sets for Monotone Recurrence Relations

2014 ◽  
Vol 24 (01) ◽  
pp. 1450012 ◽  
Author(s):  
Ya-Nan Wang ◽  
Wen-Xin Qin
Keyword(s):  
Rotation Number ◽  
Recurrence Relations ◽  
Irrational Rotation ◽  
Necessary Condition ◽  
Irrational Rotation Number

In this paper, we show that a necessary condition for nonminimal Aubry–Mather sets of monotone recurrence relations is that the set of all Birkhoff minimizers with some irrational rotation number does not constitute a foliation, i.e. the gaps of the minimal Aubry–Mather set are not filled up with Birkhoff minimizers.

scholarly journals On the structure of the family of Cherry fields on the torus

1985 ◽  
Vol 5 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Colin Boyd
Keyword(s):  
Vector Field ◽  
Hausdorff Dimension ◽  
Rotation Number ◽  
Vector Fields ◽  
Cantor Set ◽  
Irrational Rotation ◽  
Codimension One ◽  
The Family ◽  
Irrational Rotation Number

AbstractA class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.

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].

scholarly journals Attractors with irrational rotation number

2011 ◽  
Vol 153 (1) ◽  
pp. 59-77 ◽  
Author(s):  
LUIS HERNÁNDEZ-CORBATO ◽  
RAFAEL ORTEGA ◽  
FRANCISCO R. RUIZ DEL PORTAL
Keyword(s):  
Fixed Point ◽  
Rotation Number ◽  
Irrational Rotation ◽  
Asymptotically Stable ◽  
Stable Fixed Point ◽  
Region Of Attraction ◽  
Prime Ends ◽  
Irrational Rotation Number

AbstractLet h: 2 → 2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Carathéodory's prime ends theory one can associate a rotation number, ρ, to h|U. We prove that any map in the above conditions and with ρ ∉ induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, ρ ∉ if and only if Per(h) = Fix(h) = {p}.

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.

Complex bounds for renormalization of critical circle maps

1999 ◽  
Vol 19 (1) ◽  
pp. 227-257 ◽  
Author(s):  
MICHAEL YAMPOLSKY
Keyword(s):  
Rotation Number ◽  
A Priori ◽  
Julia Sets ◽  
Irrational Rotation ◽  
Local Connectivity ◽  
Circle Maps ◽  
Critical Circle ◽  
Irrational Rotation Number

We use the methods developed with Lyubich for proving complex bounds for real quadratics to extend de Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows.As another application of our methods we present a new proof of a theorem of Petersen on local connectivity of some Siegel Julia sets.

scholarly journals Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

2020 ◽  
Vol 16 (4) ◽  
pp. 651-672
Author(s):  
B. Ndawa Tangue ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Rotation Number ◽  
Critical Exponents ◽  
Irrational Rotation ◽  
Circle Maps ◽  
Irrational Rotation Number ◽  
Nonwandering Set ◽  
Flat Piece

We consider order-preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(l_1, l_2)$. We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1\backslash \bigcup^\infty_{i=0}f^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

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