scholarly journals On non-contractible periodic orbits for surface homeomorphisms

2015 ◽  
Vol 36 (5) ◽  
pp. 1644-1655 ◽  
Author(s):  
FÁBIO ARMANDO TAL
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Closed Subset ◽  
Universal Covering ◽  
Periodic Points ◽  
Uniform Bound ◽  
Large Period ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Orientable Surfaces

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if ${\hat{g}}$ is its lift to the universal covering of $S$ that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) the set of fixed points for ${\hat{g}}$ projects to a closed subset $F$ which contains an essential continuum; (2) $g$ has non-contractible periodic points of every sufficiently large period; or (3) there exists a uniform bound $M>0$ such that, if $\hat{x}$ projects to a contractible periodic point, then the ${\hat{g}}$ orbit of $\hat{x}$ has diameter less than or equal to $M$. Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.

scholarly journals Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2293-2333
Author(s):  
JINGZHI YAN
Keyword(s):  
Fixed Point ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Dirac Measure ◽  
Lefschetz Index ◽  
Weak Star ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Area Preserving ◽  
Measure Sense

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_{0}$ with Lefschetz index one. Le Roux conjectured that $z_{0}$ is accumulated by periodic orbits. In this paper, we will approach Le Roux’s conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_{0}$ and if the area of $M$ is finite, then $z_{0}$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_{0}$ is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.

Periodic orbits of continuous mappings of the circle without fixed points

1981 ◽  
Vol 1 (4) ◽  
pp. 413-417 ◽  
Author(s):  
Chris Bernhardt
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Continuous Map ◽  
Continuous Mappings ◽  
Positive Integers

AbstractLet f be a continuous map of the circle to itself. Let P(f) denote the set of periods of the periodic points. In this paper the set P(f) is studied for functions without fixed points, so 1∉P(f). In particular, it is shown that if s, t are the two smallest integers in P(f) and s and t are relatively prime then αs+βt∈P(f) for any positive integers α and β.

scholarly journals Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 101
Author(s):  
Alicia Cordero ◽  
Marlon Moscoso-Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Convergence Properties ◽  
Parameter Spaces ◽  
Numerical Tests ◽  
The Family ◽  
Dynamics Stability

In this paper, we present a new parametric family of three-step iterative for solving nonlinear equations. First, we design a fourth-order triparametric family that, by holding only one of its parameters, we get to accelerate its convergence and finally obtain a sixth-order uniparametric family. With this last family, we study its convergence, its complex dynamics (stability), and its numerical behavior. The parameter spaces and dynamical planes are presented showing the complexity of the family. From the parameter spaces, we have been able to determine different members of the family that have bad convergence properties, as attracting periodic orbits and attracting strange fixed points appear in their dynamical planes. Moreover, this same study has allowed us to detect family members with especially stable behavior and suitable for solving practical problems. Several numerical tests are performed to illustrate the efficiency and stability of the presented family.

CONTROL OF n-SCROLL CHUA'S CIRCUIT

2009 ◽  
Vol 19 (11) ◽  
pp. 3813-3822 ◽  
Author(s):  
ABDELKRIM BOUKABOU ◽  
BILEL SAYOUD ◽  
HAMZA BOUMAIZA ◽  
NOURA MANSOURI
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Predictive Control ◽  
Control Method ◽  
Sufficient Conditions ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit ◽  
Necessary And Sufficient

This paper addresses the control of unstable fixed points and unstable periodic orbits of the n-scroll Chua's circuit. In a first step, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed predictive control method. In addition, we show how a chaotic system with multiple unstable periodic orbits can be stabilized by taking the system dynamics from one UPO to another. Control performances of these approaches are demonstrated by numerical simulations.

Piecewise isometries, uniform distribution and 3log 2−π2/8

2011 ◽  
Vol 32 (6) ◽  
pp. 1862-1888 ◽  
Author(s):  
YITWAH CHEUNG ◽  
AREK GOETZ ◽  
ANTHONY QUAS
Keyword(s):  
Phase Space ◽  
Uniform Distribution ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Full Measure ◽  
Infinite Measure ◽  
Important Family ◽  
Large Numbers ◽  
Piecewise Isometries ◽  
Measure Preserving Transformations

AbstractWe use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

Periodic orbits analysis for the Zhou system via variational approach

2019 ◽  
Vol 33 (19) ◽  
pp. 1950212 ◽  
Author(s):  
Chengwei Dong ◽  
Lian Jia
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Variational Approach ◽  
Topological Classification ◽  
Dissipative Systems ◽  
Systematic Calculation ◽  
Low Dimensional ◽  
General Method

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

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