Global stability of a class of dincontinuous area-preserving maps

AbstractThe requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

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On reversible maps and symmetric periodic points

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.71 ◽

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.

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The Multiplicity of Bifurcations for Area-Preserving Maps

Bulletin of the London Mathematical Society ◽

10.1112/blms/26.4.382 ◽

1994 ◽

Vol 26 (4) ◽

pp. 382-394

Author(s):

Robert S. Mackay ◽

Tony Shardlow

Keyword(s):

Area Preserving Maps ◽

Area Preserving

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