scholarly journals On forward iterated Hausdorffness and development of embryo from zygote in bitopological dynamical systems

2021 ◽  
Vol 13(62) (2) ◽  
pp. 399-410
Author(s):  
Santanu Acharjee ◽  
Kabindra Goswami ◽  
Hemanta Kumar Sarmah
Keyword(s):  
Dynamical System ◽  
Cell Division ◽  
Dynamical Systems ◽  
Topological Space ◽  
Development Process ◽  
Hausdorff Space ◽  
Topological Dynamical System ◽  
Relative Topology ◽  
Dynamical Properties

Topological dynamical system is an area of dynamical system to investigate dynamical properties in terms of a topological space. Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] applied topological dynamical system to explore the development process of an embryo from the zygote until birth and made three conjectures. In this paper, we disprove conjecture 3 of Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] by applying some of our mathematical results of bitopological dynamical system. Also, we introduce forward iterated Hausdorff space, backward iterated Hausdorff space, pairwise iterated Hausdor_ space and establish relations between them in bitopological dynamical system. We formulate the function that represents cell division (specially, mitosis) and using this function we show that in the development process of a human baby from the zygote until its birth, there is a stage where the developing stage is forward iterated Hausdorff

Dynamics of Neural Networks as Nonlinear Systems with Several Equilibria

2011 ◽  
pp. 331-357 ◽  
Author(s):  
Daniela Danciu
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Time Delay ◽  
Dynamical Systems ◽  
Qualitative Properties ◽  
The Neural Network ◽  
The Neural Networks ◽  
Global Asymptotics ◽  
Dynamical Properties

Neural networks—both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. The chapter deals with the second kind of dynamics. More precisely, since the emergent computational capabilities of a recurrent neural network can be achieved provided it has suitable dynamical properties when viewed as a system with several equilibria, the chapter deals with those qualitative properties connected to the achievement of such dynamical properties as global asymptotics and gradient-like behavior. In the case of the neural networks with delays, these aspects are reformulated in accordance with the state of the art of the theory of time delay dynamical systems.

scholarly journals Transitive maps in bitopological dynamical systems

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2011-2021
Author(s):  
Santanu Acharjee ◽  
Kabindra Goswami ◽  
Hemanta Sarmah
Keyword(s):  
Dynamical Systems ◽  
Human Embryo ◽  
Development Process ◽  
Biological Application ◽  
Relative Topology

This paper introduces fundamental ideas of bitopological dynamical systems. Here, notions of bitopological transitivity, point transitivity, pairwise iterated compactness, weakly bitopological transitivity, etc. are introduced. Later, it is shown that under pairwise homeomorphism, weakly point transitivity implies weakly bitopological transitivity. Moreover, under pairwise homeomorphism; pairwise compactness and pairwise iterated compactness are found to be equivalent. Later, we apply our results in the development process of a human embryo from the zygote until birth. During the process of biological application, we disprove conjecture 1 of Nada and Zohny [S. I. Nada, H. Zohny, An application of relative topology in biology. Chaos, Solitons and Fractals, 42 (2009) 202-204].

scholarly journals Neutrosophic Orbit Topological Spaces

2021 ◽  
Vol 18 (24) ◽  
pp. 1443
Author(s):  
T Madhumathi ◽  
F NirmalaIrudayam
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Space ◽  
Necessary Conditions ◽  
Neutrosophic Set ◽  
Topological Spaces ◽  
Open Set ◽  
Evolution Function

Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. In the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. Hence in this paper we focus on introducing the concept of neutrosophic orbit topological space denoted as (X, tNO). Also, some of the important characteristics of neutrosophic orbit open sets are discussed with suitable examples. HIGHLIGHTS The orbit in mathematics has an important role in the study of dynamical systems Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. We combine the above two topics and create the following new concept The collection of all neutrosophic orbit open sets under the mapping . we introduce the necessary conditions on the mapping 𝒇 in order to obtain a fixed orbit of a neutrosophic set (i.e., 𝒇(𝝁) = 𝝁) for any neutrosophic orbit open set 𝝁 under the mapping 𝒇

scholarly journals Bounded complexity, mean equicontinuity and discrete spectrum

2019 ◽  
Vol 41 (2) ◽  
pp. 494-533 ◽  
Author(s):  
WEN HUANG ◽  
JIAN LI ◽  
JEAN-PAUL THOUVENOT ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Invariant Measure ◽  
Discrete Spectrum ◽  
Ergodic Theory ◽  
Topological Dynamics ◽  
Topological Dynamical System ◽  
The Mean ◽  
Bounded Complexity

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

scholarly journals Repelling invariant curves in planar discrete dynamical systems

1994 ◽  
Vol 49 (3) ◽  
pp. 469-481 ◽  
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Continuous Dependence ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Dynamical Properties ◽  
Attracting Fixed Point

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

Approximations of Dynamical Systems and Their Applications to Cryptography

2003 ◽  
Vol 13 (07) ◽  
pp. 1937-1948 ◽  
Author(s):  
J. M. Amigó ◽  
J. Szczepański
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Block Structure ◽  
Point Of View ◽  
Interval Exchange ◽  
Linear Cryptanalysis ◽  
New Approach ◽  
Interval Exchange Transformations ◽  
Measure Spaces ◽  
Dynamical Properties

During the last years a new approach to construct safe block and stream ciphers has been developed using the theory of dynamical systems. Since a block cryptosystem is generally, from the mathematical point of view, a family (parametrized by the keys) of permutations of n-bit numbers, one of the main problems of this approach is to adapt the dynamics defined by a map f to the block structure of the cryptosystem. In this paper we propose a method based on the approximation of f by periodic maps Tn (v.g. some interval exchange transformations). The approximation of automorphisms of measure spaces by periodic automorphisms was introduced by Halmos and Rohlin. One important aspect studied in our paper is the relation between the dynamical properties of the map f (say, ergodicity or mixing) and the immunity of the resulting cipher to cryptolinear attacks, which is currently one of the standard benchmarks for cryptosystems to be considered secure. Linear cryptanalysis, first proposed by M. Matsui, exploits some statistical inhomogeneities of expressions called linear approximations for a given cipher. Our paper quantifies immunity to cryptolinear attacks in terms of the approximation speed of the map f by the periodic Tn. We show that the most resistant block ciphers are expected when the approximated dynamical system is mixing.

scholarly journals Characteristic measures of symbolic dynamical systems

2021 ◽  
pp. 1-7
Author(s):  
JOSHUA FRISCH ◽  
OMER TAMUZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Automorphism Group ◽  
Probability Measure ◽  
Automorphism Groups ◽  
Topological Dynamical System ◽  
Zero Entropy ◽  
Characteristic Measure

Abstract A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

scholarly journals The structure of tame minimal dynamical systems

2007 ◽  
Vol 27 (6) ◽  
pp. 1819-1837 ◽  
Author(s):  
ELI GLASNER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Probability Measure ◽  
Topological Space ◽  
Haar Measure ◽  
Invariant Probability Measure ◽  
Structure Theory ◽  
Convergence Of Sequences ◽  
Minimal Dynamical Systems ◽  
Minimal Dynamical System

AbstractA dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

scholarly journals Simple dynamical systems

2019 ◽  
Vol 20 (2) ◽  
pp. 307 ◽  
Author(s):  
K. Ali Akbar ◽  
V. Kannan ◽  
I. Subramania Pillai
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Equivalence Class ◽  
Equivalence Relation ◽  
Title Page ◽  
The Family ◽  
Dynamical Properties ◽  
Simple Systems

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>In this paper, we study the class of simple systems on </span><span>R </span><span>induced by homeomorphisms having finitely many non-ordinary points. We characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. For </span><span>x,y </span><span>∈ </span><span>R</span><span>, we say </span><span>x </span><span>∼ </span><span>y </span><span>on a dynamical system (</span><span>R</span><span>,f</span><span>) if </span><span>x </span><span>and </span><span>y </span><span>have same dynamical properties, which is an equivalence relation. Said precisely, </span><span>x </span><span>∼ </span><span>y </span><span>if there exists an increasing homeomorphism </span><span>h </span><span>: </span><span>R </span><span>→ </span><span>R </span><span>such that </span><span>h </span><span>◦ </span><span>f </span><span>= </span><span>f </span><span>◦ </span><span>h </span><span>and </span><span>h</span><span>(</span><span>x</span><span>) = </span><span>y</span><span>. </span><span>An element </span><span>x </span><span>∈ </span><span>R </span><span>is </span><span>ordinary </span><span>in (</span><span>R</span><span>,f</span><span>) if its equivalence class [</span><span>x</span><span>] is a neighbourhood of it.</span></p><p><span><br /></span></p></div></div></div>

Sign in / Sign up

Export Citation Format

Share Document