scholarly journals Furstenberg Families and Sensitivity

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Huoyun Wang ◽  
Jincheng Xiong ◽  
Feng Tan
Keyword(s):  
Dynamical System ◽  
Open Neighborhood ◽  
Furstenberg Family

We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system(X,f)isℱ-sensitive if there exists a positiveεsuch that for everyx∈Xand every open neighborhoodUofxthere existsy∈Usuch that the pair(x,y)is notℱ-ε-asymptotic; that is, the time set{n:d(fn(x),fn(y))>ε}belongs toℱ, whereℱis a Furstenberg family. A dynamical system(X,f)is (ℱ1,ℱ2)-sensitive if there is a positiveεsuch that everyx∈Xis a limit of pointsy∈Xsuch that the pair(x,y)isℱ1-proximal but notℱ2-ε-asymptotic; that is, the time set{n:d(fn(x),fn(y))<δ}belongs toℱ1for any positiveδbut the time set{n:d(fn(x),fn(y))>ε}belongs toℱ2, whereℱ1andℱ2are Furstenberg families.

scholarly journals Point transitivity, -transitivity and multi-minimality

2014 ◽  
Vol 35 (5) ◽  
pp. 1423-1442 ◽  
Author(s):  
ZHIJING CHEN ◽  
JIAN LI ◽  
JIE LÜ
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Topological Dynamical System ◽  
Property A ◽  
Weak Mixing ◽  
Furstenberg Family ◽  
Strongly Mixing ◽  
Transitive Point ◽  
Weakly Mixing

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

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