scholarly journals A Topological Approach to Bend-Twist Maps with Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Anna Pascoletti ◽  
Fabio Zanolin
Keyword(s):  
Fixed Point Theorems ◽  
Small Perturbations ◽  
Topological Approach ◽  
Twist Map ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Planar Systems ◽  
Twist Theorem ◽  
Area Preserving

In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

scholarly journals On reversible maps and symmetric periodic points

2016 ◽  
Vol 38 (4) ◽  
pp. 1479-1498
Author(s):  
JUNGSOO KANG
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Periodic Points ◽  
Analogous Statement ◽  
Twist Map ◽  
Planar Domains ◽  
Area Preserving Maps ◽  
Symmetric Periodic Orbits ◽  
Area Preserving ◽  
Double Symmetries

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.

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.

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.

scholarly journals Hamiltonian Maps for Heliac Magnetic Islands

1995 ◽  
Vol 48 (5) ◽  
pp. 871 ◽  
Author(s):  
MG Davidson ◽  
RL Dewar ◽  
HJ Gardner ◽  
J Howard
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Chaos Theory ◽  
Magnetic Islands ◽  
Twist Map ◽  
Hamiltonian Chaos ◽  
Area Preserving Maps ◽  
Vacuum Fields ◽  
Rotational Transform ◽  
Area Preserving

Magnetic islands in toroidal heliac stellarator vacuum fields are explored with Hamiltonian chaos theory and the associated area-preserving maps. Magnetic field line island chains are examined first analytically, with perturbation theory, and then numerically to produce Poincaré sections, which are compared with H−1 Heliac stellarator puncture plot diagrams. Rotational transform profiles are chosen to permit the comparison of twist map and nontwist map predictions with field line behaviour computed by a field line tracing computer program and observed experimentally.

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 Hyperbolic sets for twist maps

1985 ◽  
Vol 5 (3) ◽  
pp. 337-339 ◽  
Author(s):  
Daniel L. Goroff
Keyword(s):  
Hyperbolic Structure ◽  
Twist Map ◽  
Twist Maps ◽  
Hyperbolic Sets ◽  
Area Preserving

AbstractAn example is given of an area-preserving monotone twist map such that a uniformly hyperbolic structure exists on the closure of its Birkhoff maximizing orbits.

scholarly journals A topological version of a theorem of Mather on twist maps

1984 ◽  
Vol 4 (4) ◽  
pp. 585-603 ◽  
Author(s):  
Glen Richard Hall
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Point ◽  
Rotation Number ◽  
Point Theorem ◽  
Twist Map ◽  
Twist Maps ◽  
Recent Theorem ◽  
Topological Version ◽  
Area Preserving

AbstractIn this report we show that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. We use topological techniques so no assumption of area-preservation or circle intersection property is needed. If the map is area-preserving then this theorem andthe fixed point theorem of Birkhoff imply a recent theorem of Aubry and Mather. We also show that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff.

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 Positive Solutions for Nonlinear Fractional Differential Equations with Boundary Conditions Involving Riemann-Stieltjes Integrals

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Jiqiang Jiang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Integral Boundary Value Problems

We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.

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