Some dynamical properties of the family of tent maps (original) (raw)
Bifurcations and Periodic Orbits in Chaotic Maps
Open Systems & Information Dynamics (OSID), 2001
A hierarchy of universalities in families of 1-D maps is discussed. Breakdown of universalities in families of 3-D maps is shown on selected examples of such families.
An Elementary Study of Chaotic Behaviors in 1-D Maps
Journal of Applied Mathematics and Physics, 2019
In this article, we have discussed basic concepts of one-dimensional maps like Cubic map, Sine map and analyzed their chaotic behaviors in several senses in the unit interval. We have mainly focused on Orbit Analysis, Time Series Analysis, Lyapunov Exponent Analysis, Sensitivity to Initial Conditions, Bifurcation Diagram, Cobweb Diagram, Histogram, Mathematical Analysis by Newton's Iteration, Trajectories and Sensitivity to Numerical Inaccuracies of the said maps. We have tried to make decision about these mentioned maps whether chaotic or not on a unique interval of parameter value. We have performed numerical calculations and graphical representations for all parameter values on that interval and have tried to find if there is any single value of parameter for which those maps are chaotic. In our calculations we have found there are many values for which those maps are chaotic. We have showed numerical calculations and graphical representations for single value of the parameter only in this paper which gives a clear visualization of chaotic dynamics. We performed all graphical activities by using Mathematica and MATLAB.
An elementary approach to dynamics and bifurcations of skew tent maps
Journal of Difference Equations and Applications, 2008
In this paper the dynamics of skew tent maps are classified in terms of two bifurcation parameters. In time series analysis such maps are usually referred to as continuous threshold autoregressive models (TAR(1) models) after . This study contains results simplifying the use of TAR(1) models considerably, e. g. if a periodic attractor exists it is unique. On the other hand we also claim that care must be exercised when threshold autoregressive (TAR) models are used. In fact, they possess a very special type of dynamical pattern with respect to the bifurcation parameters and their transition to chaos is far from standard.
The discontinuous flat top tent map and the nested period incrementing bifurcation structure
Chaos, Solitons & Fractals, 2012
In this work we report the recently discovered nested period incrementing bifurcation scenario. The investigated piecewise linear map is defined on three partitions of the unit interval, constant in the middle partition and therefore displays a rich variety of superstable orbits. These orbits are arranged according to an infinite binary tree of the corresponding symbolic sequences, which can be generated by a simple set of rules. The system also allows for straightforward computation of the respective regions of existence. One of the most striking results of our investigations is that the famous U-sequence is inevitably embedded in the nested period incrementing scenario.
Applications of Bifurcation and Chaos on Discrete Time Dynamical System
International Journal of Computer Applications
In this paper the study of rigorous basic dynamical facts on bifurcation and chaos for discrete models in time dynamics and introduce a generalized logistic map and its dynamical behavior with tent and Henon Map has recognized.Different discrete curves have been developed and more general biological logistic curve are studied. Review and compare several such maps and analysis properties of those maps on the applications of bifurcation and chaos. Discuss the concept of chaos and bifurcations in the discrete time dynamical tent maps and generalized logistic growth models as time dynamical attractor.
Bifurcation Analysis of Periodic Orbits of Maps in Matlab
We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a matlab toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., LP, PD and NS), and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are i...
On the dynamics of the Tent function - Phase diagrams
Journal of Advanced Studies in Topology, 2016
This paper focuses on the study of a one-dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.
Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map
ESAIM: Proceedings, 2012
In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns out that there exists a strong connection between stable and unstable orbits and that the appearance of a single unstable orbit may cause an infinite number of stable orbits to appear. Based on this connection we provide a more precise explanation of the recently discovered self-similar bifurcation scenario occurring in the discontinuous flat top tent map denoted as the nested period incrementing scenario.
Chaos in the Dynamics of the Family of Mappings f c (x) = x 2 – x + c
In this paper, we will study the chaotic behaviour of the family of quadratic mappings f c (x) = x 2 -x + c through its dynamics. In first few sections, we will take a review of some basic definitions and examples including a dynamical system, orbit, fixed and periodic, etc. Later, we will prove some results that analyse the nature and the stability of the fixed and periodic points of a dynamical system. Using these results, we will study the dynamics of the family of mappings f c (x) = x 2 -x + c for various values of the real constant c.
Abstract and Applied Analysis
The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and cha...
DETERMINATION OF PERIOD-3 CYCLE VIA SYLVESTER MATRIX IN THE CHAOTIC REGION OF A ONE DIMENSIONAL MAP
IASET, 2013
In science, for a long time, it has been assumed that regularity therefore predictability has been the centre of approaches to explain the behaviours of systems. Whereas in real life, it is a well known fact that systems exhibit unexpected behaviours which lead to irregular and unpredictable outcomes. This approach, named as non-linear dynamics, produces much closer representation of real happenings. The chaos theory which is one of methods of non-linear dynamics, has recently attracted many scientist from all different fields. In this paper we analyse a situation in which the sequence { } is non-periodic and might be called "chaotic”. Here we have considered a one parameter map (Verhulst population model), obtained the parameter value 𝛌for which period-3 cycle is created in a Tangent bifurcation, using Sarkovskii‟s Theorem, Sylvester‟s Matrix and Resultant. We also calculated the parameter range 𝛌0<𝛌<𝛌1 for which the map possesses stable period-3 orbit.
Several Chaotic Approaches of One Dimensional Doubling Map
2018
In this paper, we study basic dynamical behavior of one-dimensional Doubling map. Especially emphasis is given on the chaotic behaviors of the said map. Several approaches of chaotic behaviors by some pioneers it is found that the Doubling map is chaotic in different senses. We mainly focused on Orbit Analysis, Sensitivity to Initial Conditions, Sensitivity to Numerical Inaccuracies, Trajectories and Staircase Diagram of the Doubling map. The graphical representations show that this map is chaotic in different senses. The behavior of the said map is found irregular, that is, chaotic.
Stability and Feigenbaum's Universality in Two Dimensional Chaotic Map 1
In this paper a two dimensional non linear map is taken whose period doubling dynamical behavior has been analyzed. The bifurcation points have been calculated numerically and have been observed that the map follows a universal behavior that has been proposed by Feigenbaum. With the help of experimental bifurcation points the accumulation point where chaos starts has been calculated.
A novel one-dimensional chaotic map with improved sine map dynamics
International Journal of Electrical and Computer Engineering (IJECE), 2025
These days, keeping information safe from people who should not have access to it is very important. Chaos maps are a critical component of encryption and security systems. The classical one-dimensional maps, such as logistic, sine, and tent, have many weaknesses. For example, these classical maps may exhibit chaotic behavior within the narrow range of the rate variable between 0 and 1and the small interval's rate variable. In recent years, several researchers have tried to overcome these problems. In this paper, we propose a new one-dimensional chaotic map that improves the sine map. We introduce an additional parameter and modify the mathematical structure to enhance the chaotic behavior and expand the interval's rate variable. We evaluate the effectiveness of our map using specific tests, including fixed points and stability analysis, Lyapunov exponent analysis, diagram bifurcation, sensitivity to initial conditions, the cobweb diagram, sample entropy and the 0-1 test.
Cycling Chaos in One-Dimensional Coupled Iterated Maps
International Journal of Bifurcation and Chaos, 2002
Cycling behavior involving steady-states and periodic solutions is known to be a generic feature of continuous dynamical systems with symmetry. Using Chua's circuit equations and Lorenz equations, Dellnitz et al. [1995] showed that "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets, can also be found generically in coupled cell systems of differential equations with symmetry. In this work, we use numerical simulations to demonstrate that cycling chaos also occurs in discrete dynamical systems modeled by one-dimensional maps. Using the cubic map f (x, λ) = λx - x3 and the standard logistic map, we show that coupled iterated maps can exhibit cycles connecting fixed points with fixed points and periodic orbits with periodic orbits, where the period can be arbitrarily high. As in the case of coupled cell systems of differential equations, we show that cycling behavior can also be a feature of the global dynamics of coupled itera...
Dynamical analysis of one two-dimensional map
2018
In this paper, we will present a dynamical analysis of a two- dimensional map via an example. The fixed points, the classification of their character (stable or unstable), the visualization of some orbits and plotting of the bifurcation diagrams will be the main aspects of research for the two-dimensional map. As computer support, mathematical software Mathematica will be used
Giga-Periodic Orbits for Weakly Coupled Tent and Logistic Discretized Maps
Simple dynamical systems often involve periodic motion. Quasi-periodic or chaotic motion is frequently present in more complicated dynamical systems. However, for the most part, underneath periodic motion models chaotic motion. Chaotic attractors are nearly always present in such dissipative systems. Since their discovery in 1963 by E. Lorenz, they have been extensively studied in order to understand their nature. In the past decade, the aim of the research has been shifted to the applications for industrial mathematics. Their importance in this field is rapidly growing. Chaotic orbits embedded in chaotic attractor can be controlled allowing the possibility to control laser beams or chemical processes and improving techniques of communications. They can also produce very long sequences of numbers which can be used as efficiently as random numbers even if they have not the same nature. However, mathematical results concerning chaotic orbits are often obtained using sets of real numbers (belonging to R or R n) (e.g. the famous theorem of A. N. Sharkovskiǐ which defines which ones periods exist for continuous functions such as logistic or tent maps). O.E. Lanford III reports the results of some computer experiments on the orbit structure of the discrete maps on a finite set which arise when an expanding map of the circle is iterated "naively" on the computer. There is a huge gap between these results and the theorem of Sharkovskiǐ, due to the discrete nature of floating points used by computers. This article introduces new models of very very weakly coupled logistic and tent maps for which orbits of very long period are found. The length of these periods is far greater than one billion. We call giga-periodic orbits such orbits for which the length is greater than 10 9 and less than 10 12. Tera, and peta periodic orbits are the name of the orbits the length of which is one thousand or one million greater. The property of these models relatively to the distribution of the iterates (invariant measure) are described. They are found very useful for industrial mathematics for a variety of purposes such as generation of cryptographic keys, computer games and some classes of scientific experiments.
Comprehensive analysis and comparison of mappings as dynamical systems can be implemented with support of mathematical software. The most commonly used mathematical software that gives good results in the theory of dynamical systems, respectively in teaching and in scientific research is “Mathematica”. In this paper we are going to use “Mathematica” for characterization and comparison of one parameter families of square mappings as dynamic systems. Their behavior will be viewed through the prism of the fixed points, finding and analyzing periodic points with period 2 and analysis of their bifurcation diagrams depending on the changing of real parameter.
ELECTRICAL AND COMPUTER SYSTEMS, 2021
In this paper we consider the processes in maps, which are examples of nonlinear dynamical systems. Analyzing dynamical systems, it is necessary to take into account and analyze properties of iterative functions that determine the length of nonrepetitive iterative process. It is shown that not only properties of functions, but also properties of numbers from the considered functions domain influence the nonlinear maps behavior.