scholarly journals Rotation sets and Morse decompositions in twist maps

1988 ◽  
Vol 8 (8) ◽  
pp. 33-61 ◽  
Keyword(s):  
Fixed Point ◽  
Main Tool ◽  
Rotation Band ◽  
Twist Maps ◽  
Alternative Hypotheses ◽  
Transitive Set ◽  
Morse Decompositions ◽  
A Chain ◽  
Rotation Set

AbstractPositive tilt maps of the annulus are studied, and a correspondence is developed between the rotation set of the map and certain of its Morse decompositions. The main tool used is a characterization of fixed point free lifts of positive tilt maps. As an application, some alternative hypotheses under which the conclusions of the Aubry-Mather theorem hold are given, and it is also shown that the rotation band of a chain transitive set is always in the rotation set of the map.

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 Hyperstability of the Fréchet Equation and a Characterization of Inner Product Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Anna Bahyrycz ◽  
Janusz Brzdęk ◽  
Magdalena Piszczek ◽  
Justyna Sikorska
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Main Tool ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Fréchet Equation ◽  
New Inequalities

We prove some stability and hyperstability results for the well-known Fréchet equation stemming from one of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces. We finish the paper with some new inequalities characterizing the inner product spaces.

scholarly journals Recurrence and fixed points of surface homeomorphisms

1988 ◽  
Vol 8 (8) ◽  
pp. 99-107 ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lebesgue Measure ◽  
Invariant Chain ◽  
Transitive Set ◽  
Surface Homeomorphisms ◽  
A Chain ◽  
Area Preserving ◽  
Twist Condition

AbstractWe prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

Characterization of seawater reverse osmosis fouled membranes from large scale commercial desalination plant

2019 ◽  
Author(s):  
Chem Int
Keyword(s):  
Reverse Osmosis ◽  
Large Scale ◽  
Retention Rate ◽  
Local Level ◽  
Microscopic Analysis ◽  
Transmembrane Pressure ◽  
Hydraulic Permeability ◽  
Seawater Reverse Osmosis ◽  
A Chain

The objective of this work is to study the ageing state of a used reverse osmosis (RO) membrane taken in Algeria from the Benisaf Water Company seawater desalination unit. The study consists of an autopsy procedure used to perform a chain of analyses on a membrane sheet. Wear of the membrane is characterized by a degradation of its performance due to a significant increase in hydraulic permeability (25%) and pressure drop as well as a decrease in salt retention (10% to 30%). In most cases the effects of ageing are little or poorly known at the local level and global measurements such as (flux, transmembrane pressure, permeate flow, retention rate, etc.) do not allow characterization. Therefore, a used RO (reverse osmosis) membrane was selected at the site to perform the membrane autopsy tests. These tests make it possible to analyze and identify the cause as well as to understand the links between performance degradation observed at the macroscopic scale and at the scale at which ageing takes place. External and internal visual observations allow seeing the state of degradation. Microscopic analysis of the used membranes surface shows the importance of fouling. In addition, quantification and identification analyses determine a high fouling rate in the used membrane whose foulants is of inorganic and organic nature. Moreover, the analyses proved the presence of a biofilm composed of protein.

A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS

1994 ◽  
Vol 04 (04) ◽  
pp. 979-998 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Main Tool ◽  
Lyapunov Stability Theory ◽  
Asymptotical Stability ◽  
Unified Framework ◽  
Partial Synchronization ◽  
Chua’S Oscillator ◽  
And Control

In this paper, we give a framework for synchronization of dynamical systems which unifies many results in synchronization and control of dynamical systems, in particular chaotic systems. We define concepts such as asymptotical synchronization, partial synchronization and synchronization error bounds. We show how asymptotical synchronization is related to asymptotical stability. The main tool we use to prove asymptotical stability and synchronization is Lyapunov stability theory. We illustrate how many previous results on synchronization and control of chaotic systems can be derived from this framework. We will also give a characterization of robustness of synchronization and show that master-slave asymptotical synchronization in Chua’s oscillator is robust.

scholarly journals Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

10.37236/6898 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Hsiang-Chun Hsu ◽  
Yu-Pei Huang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Rectangular Shape ◽  
Explicit Characterization ◽  
Finite Poset ◽  
Maximal Chains

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.

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