On Two-Dimensional Area-Preserving Maps with Homoclinic Tangencies that Have Infinitely Many Generic Elliptic Periodic Points

2005 ◽  
Vol 128 (2) ◽  
pp. 2767-2773 ◽  
Author(s):  
S. V. Gonchenko ◽  
L. P. Shilnikov
Keyword(s):  
Periodic Points ◽  
Two Dimensional ◽  
Area Preserving Maps ◽  
Homoclinic Tangencies ◽  
Area Preserving

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.

scholarly journals On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

Nonlinearity ◽  
2015 ◽  
Vol 28 (9) ◽  
pp. 3027-3071 ◽  
Author(s):  
Amadeu Delshams ◽  
Marina Gonchenko ◽  
Sergey Gonchenko
Keyword(s):  
Area Preserving Maps ◽  
Homoclinic Tangencies ◽  
Area Preserving

Corrigendum: On dynamics and bifurcations of area-preserving maps with homoclinic tangencies (2015 Nonlinearity 28 3027)

Nonlinearity ◽  
2017 ◽  
Vol 30 (3) ◽  
pp. C2-C2
Author(s):  
Amadeu Delshams ◽  
Marina Gonchenko ◽  
Sergey Gonchenko
Keyword(s):  
Area Preserving Maps ◽  
Homoclinic Tangencies ◽  
Area Preserving

scholarly journals A symplectic proof of a theorem of Franks

2012 ◽  
Vol 148 (6) ◽  
pp. 1969-1984 ◽  
Author(s):  
Brian Collier ◽  
Ely Kerman ◽  
Benjamin M. Reiniger ◽  
Bolor Turmunkh ◽  
Andrew Zimmer
Keyword(s):  
Crucial Role ◽  
Additional Assumption ◽  
Periodic Points ◽  
Two Dimensional ◽  
Symplectic Topology ◽  
Hamiltonian Diffeomorphisms ◽  
Area Preserving

AbstractA celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area-preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we re-prove Franks’ theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorphisms.

scholarly journals Feigenbaum's ratios of two-dimensional area preserving maps

1980 ◽  
Vol 80 (4) ◽  
pp. 217-219 ◽  
Author(s):  
B. Derrida ◽  
Y. Pomeau
Keyword(s):  
Two Dimensional ◽  
Area Preserving Maps ◽  
Area Preserving

MULTIPLIERS OF HOMOCLINIC TANGENCIES AND A THEOREM OF GONCHENKO AND SHILNIKOV ON AREA PRESERVING MAPS

2005 ◽  
Vol 15 (11) ◽  
pp. 3589-3594 ◽  
Author(s):  
VALENTIN AFRAIMOVICH ◽  
TODD YOUNG
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Set ◽  
Differential Invariants ◽  
Invariant Sets ◽  
Area Preserving Maps ◽  
Homoclinic Tangencies ◽  
Area Preserving

We investigate differential invariants for homoclinic tangencies and discuss the role of these invariants in the Hausdorff dimension of invariant sets associated with the tangency and its unfoldings. Further, we give a streamlined proof of a theorem of Goncheko and Shilnikov on the case of a tangency in an area preserving map. Specifically, the invariants determine whether or not a hyperbolic invariant set is formed near the tangency.

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