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 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.

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.

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 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 Monotone Flows with Dense Periodic Orbits

2019 ◽  
Vol 12 (4) ◽  
pp. 1350-1359
Author(s):  
Morris W. Hirsch
Keyword(s):  
Periodic Orbits ◽  
Convex Cone ◽  
Periodic Points ◽  
Nonempty Interior ◽  
Closed Convex Cone ◽  
Partially Ordered ◽  
Open Set

Let R d be partially ordered by a closed convex cone K ⊂ R d having nonempty interior: y x ⇐⇒ y − x ∈ K. Assume X ⊂ R d is a connected open set, and ϕ a flow on X that is monotone for this order: If y x and t ≥ 0, then ϕ t y ϕ t y. Theorem: If periodic points are dense, ϕ is globally periodic.

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 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.

Translator and translation research: general descriptions of concepts

2019 ◽  
Author(s):  
Darian Jancowicz-Pitel
Keyword(s):  
Quantitative Method ◽  
Point Of View ◽  
Translation Process ◽  
Translation Research ◽  
Qualitative And Quantitative ◽  
The Third ◽  
Part Time ◽  
Set Up ◽  
Translation Errors ◽  
Multiple Languages

The presented paper aimed for exploring the translation process, a translator or interpreter needs equipment or tools so that the objectives of a translation can be achieved. If an interpreter needs a pencil, paper, headphones, and a mic, then an interpreter needs even more tools. The tools required include conventional and modern tools. Meanwhile, the approach needed in research on translation is qualitative and quantitative, depending on the research objectives. If you want to find a correlation between a translator's translation experience with the quality or type of translation errors, a quantitative method is needed. Also, this method is very appropriate to be used in research in the scope of teaching translation, for example from the student's point of view, their level of intelligence regarding the quality or translation errors. While the next method is used if the research contains translation errors, procedures, etc., it is more appropriate to use qualitative methods. Seeing this fact, these part-time translators can switch to the third type of translator, namely free translators. This is because there is an awareness that they can live by translation. These translators set up their translation efforts that involve multiple languages.

Rail-to-Earth Resistance Assessment for a Medium Capacity Transit System With Continuous Negative Rails by Potential Measurement

2016 ◽  
Author(s):  
Bih-Yuan Ku ◽  
Ching Liang Wu ◽  
Chun-Fu Lin
Keyword(s):  
Unit Length ◽  
Point Of View ◽  
Voltage Sag ◽  
Rapid Transit ◽  
Transit System ◽  
Potential Measurement ◽  
Qualitative And Quantitative ◽  
Ground Resistance ◽  
Traction Current ◽  
Resistance Assessment

This paper presents the development of a qualitative and quantitative assessment of the resistance to ground for the electrically continuous negative rails of a medium capacity transit line of the Taipei Rapid Transit System. Using synchronous potential measurements at three stations we examine potential profiles to locate potential rail sections with low resistance to ground qualitatively. Also the voltage sag values are used to quantitatively calculate rail-to-ground resistance per unit length. The approach presented in this paper requires only voltage measurements with the traction current as the energization source. Thus, this approach can be performed as a routine maintenance procedure to obtain rail-to-ground resistance values from a system-wide point of view.

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