scholarly journals Chaotic difference equations are dense

1976 ◽  
Vol 15 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Peter E. Kloeden
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Difference Equations ◽  
Real Line ◽  
Dense Subset ◽  
Structural Instability ◽  
Compact Interval ◽  
The Real ◽  
Continuous Mappings ◽  
The Difference

A continuous function mapping a compact interval of the real line into itself is called chaotic if the difference equation defined in terms of it behaves chaotically in the sense of Li and Yorke. The set of all such chaotic functions is shown to toe a dense subset of the space of continuous mappings of that interval into itself with the max norm. This result indicates the structural instability of nonchaotic difference equations with respect to chaotic behaviour.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

scholarly journals A Remark on Isometries of Absolutely Continuous Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-3
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Continuous Function ◽  
Composition Operator ◽  
Real Line ◽  
Weighted Composition Operator ◽  
Function Spaces ◽  
Absolutely Continuous ◽  
The Real ◽  
Absolutely Continuous Function ◽  
Weighted Composition ◽  
Vector Valued

The purpose of this article is to study the isometries between vector-valued absolutely continuous function spaces, over compact subsets of the real line. Indeed, under certain conditions, it is shown that such isometries can be represented as a weighted composition operator.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

Dynamics of a Higher Order Rational Difference Equation

2011 ◽  
Vol 216 ◽  
pp. 50-55 ◽  
Author(s):  
Yi Yang ◽  
Fei Bao Lv
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we address the difference equation xn=pxn-s+xn-t/q+xn-t n=0,1,... with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0. Our results not only include some previously known results, but apply to some difference equations that have not been investigated so far.

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
Image Position

1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

scholarly journals Interval Model of the Efficiency of the Functioning of Information Web Resources for Services on Ecological Expertise

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2116
Author(s):  
Mykola Dyvak ◽  
Oleksandr Papa ◽  
Andrii Melnyk ◽  
Andriy Pukas ◽  
Nataliya Porplytsya ◽  
...  
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Model Building ◽  
Experimental Studies ◽  
Discrete Models ◽  
Model Parameters ◽  
Incorrect Choice ◽  
Web Resources ◽  
Web Resource ◽  
The Difference

Mathematical models of the efficiency dynamics of information web resources are considered in this paper. The application of interval discrete models in the form of difference equations is substantiated and the approach to estimation of the model parameters is proposed. The proposed approach is based on the artificial bee colony algorithm (ABCA). A number of experimental studies have been carried out based on data on the functioning of web resources related to environmental monitoring services. The indicator of an information web resource user’s activity has been investigated. Three cases of model building in the form of difference equations as interval discrete models (IDM) have been considered. They vary in the general kind of expression. As a result of the computational experiments, it is shown that the adequacy of a model depends on the expression of the difference equation. In the case of its incorrect choice, the proposed method of parameters’ identification may be ineffective. The obtained interval discrete model in the difference equation form, which describes the efficiency of a web resource, makes it possible to optimize business processes in an organization that uses this web resource, as well as optimally allocate organizational resources and the workload of employees of the administrative service center. Based on the conducted experiments, the efficiency of the proposed model’s application is confirmed.

scholarly journals Continuous functions of bounded nth variation

1969 ◽  
Vol 16 (3) ◽  
pp. 205-214
Author(s):  
Gavin Brown
Keyword(s):  
Continuous Function ◽  
Positive Integer ◽  
Decomposition Theorem ◽  
Continuous Functions ◽  
Unit Interval ◽  
Jordan Decomposition ◽  
The Real ◽  
Elementary Construction ◽  
The Difference

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

Entropy of general diffeomorphisms on line

2020 ◽  
pp. 1-17
Author(s):  
BAOLIN HE
Keyword(s):  
Topological Entropy ◽  
Real Line ◽  
Dense Subset ◽  
The Real ◽  
First Derivative ◽  
Hyperbolic Diffeomorphisms ◽  
On Line ◽  
Real Analytic

We study the continuity of topological entropy of general diffeomorphisms on line. First, we prove that the entropy map is continuous with respect to the strong $C^{0}$ -topology on the union of uniformly topologically hyperbolic diffeomorphisms contained in $\text{Diff}_{0}^{r}(\mathbb{R})$ (whose first derivative is uniformly away from zero), which is a $C^{0}$ -open and $C^{r}$ -dense subset of $\text{Diff}_{0}^{r}(\mathbb{R})$ , $r=1,2,\ldots ,\infty$ , and $\unicode[STIX]{x1D714}$ (real analytic). Second, we give some examples where entropy map is not continuous. Finally, we prove some results on the continuity of entropy of general diffeomorphisms on the (real) line.

Images of Bernstein sets via continuous functions

2019 ◽  
Vol 26 (4) ◽  
pp. 499-503
Author(s):  
Jacek Cichoń ◽  
Michał Morayne ◽  
Robert Rałowski
Keyword(s):  
Continuous Function ◽  
Real Line ◽  
Continuous Functions ◽  
Positive Answer ◽  
Continuous Mappings

Abstract We examine images of Bernstein sets via continuous mappings. Among other results, we prove that there exists a continuous function {f\colon\mathbb{R}\to\mathbb{R}} that maps every Bernstein subset of {\mathbb{R}} onto the whole real line. This gives the positive answer to a question of Osipov.

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