An Elementary Study of Chaotic Behaviors in 1-D Maps

Physical Review A, 1989

An analytical study of the dynamics of a piecewise cubic map depending on two parameters is carried out in this paper. For this type of map, it is shown that a stable fixed point coexists with a chaotic attractor. The way in which the deterministic dynamics of the map undergoes chaos is not representable by means of the standardly proposed routes to chaos. It appears that in this case the chaos is initiated by the appearance of an unstable solution born out of a tangent bifurcation. The general geometric approach presented here in obtaining the analytical results does not make use of the Schwarzian curvature of the map. Moreover, it is shown that only the first few iterates essentially describe the global dynamics of the map. The methodology presented here could be used to good advantage for understanding the dynamics of other types of maps as well. Computational results are provided by the cell-mapping method to expose and confirm the analytical results given in this paper.

The objective of this paper is to present a suite of applications that allow the simulation and study of chaotic systems, as well as the estimation of the most important properties associated with them. These applications implement fundamental algorithms from the field of chaotic system dynamics, such as the reconstruction of the system trajectory in the appropriate embedding space, and the estimation of the Lyapunov exponents and the fractal dimension. Furthermore, they provide additional features such as the study of bifurcation diagrams and the detection of chaotic regions in the parameter space. The current version of the applications has been developed in the programming framework of Visual C++ 6.0 and they can be used under the operating system of Microsoft Windows.

Pramana-journal of Physics, 1992

This paper is a review of the present status of studies relating to occurrence of deterministic chaos and its characterization in one-dimensional maps. As our primary aim is to introduce the nonspecialists into this fascinating world of chaos we start from very elementary concepts and give sufficient arguments for clarity of ideas. The two main scenarios during onset of chaos viz. the period doubling and intermittency are dealt with in detail. Although the logistic map is often discussed by way of illustration, a few more interesting maps are mentioned towards the end.

Abstract|This paper reports a method to dissect the orbit structure of quantized chaotic maps of the unit interval. The¯nite precision of computer arithmetic yields a spatially discrete dynamical system whose behavior is quite di®erent from that expected on the continuum of real numbers. All computed orbits are eventually periodic; which is in stark contrast to the theoretical dynamics on the real line. The dynamical behavior of quantized 1-D maps of the unit interval is characterized in terms of a) The period and quantity of the cycles where every orbit eventually lands after ā nite number of iterations of the map; and b) The length and quantity of the paths that lead to these orbits. An e±cient algorithm to compute these descriptors is proposed. The simulation results are theoretically justi¯ed in particular cases.

Journal of Software Engineering and Applications, 2015

Chaos theory attempts to explain the result of a system that is sensitive to initial conditions, complex, and shows an unpredictable behaviour. Chaotic systems are sensitive to any change or changes in the initial condition(s) and are unpredictable in the long term. Chaos theory are implementing today in many different fields of studies. In this research, we propose a new one-dimensional Triangular Chaotic Map (TCM) with full intensive chaotic population. TCM chaotic map is a one-way function that prevents the finding of a relationship between the successive output values and increases the randomness of output results. The tests and analysis results of the proposed triangular chaotic map show a great sensitivity to initial conditions, have unpredictability, are uniformly distributed and random-like and have an infinite range of intensive chaotic population with large positive Lyapunov exponent values. Moreover, TCM characteristics are very promising for possible utilization in many different study fields.

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.

Applied Sciences, 2021

Over the past decade, chaotic systems have found their immense application in different fields, which has led to various generalized, novel, and modified chaotic systems. In this paper, the general jerk equation is combined with a scaled sine map, which has been approximated in terms of a polynomial using Taylor series expansion for exhibiting chaotic behavior. The paper is based on numerical simulation and experimental verification of the system with four control parameters. The proposed system’s chaotic behavior is verified by calculating different chaotic invariants using MATLAB, such as bifurcation diagram, 2-D attractor, Fourier spectra, correlation dimension, and Maximum Lyapunov Exponent (MLE). Experimental verification of the system was carried out using Op-Amps with analog multipliers.

Journal of Statistical Physics, 1984

We consider single-humped symmetric one-dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one. To calculate the probability distribution function, and in turn the Lyapunov exponent and the correlation function, a perturbation expansion is developed for the invariant measure. Besides deriving some general results, we treat several examples in detail and compare our theoretical results with recent numerical ones. Furthermore, a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes.

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.

Chaos, Solitons & Fractals, 2001

In the present paper we study certain characteristic features associated with bifurcations of chaos in a ®nite dimensional dynamical system ± Murali±Lakshmanan±Chua (MLC) circuit equation and an in®nite dimensional dynamical system ± one-way coupled map lattice (OCML) system. We characterize chaotic attractors at various bifurcations in terms of r n q ± the variance of¯uctuations of coarse-grained local expansion rates of nearby orbits. For all chaotic attractors the r n q versus q plot exhibits a peak at q q a. Additional peaks, however, are found only just before and just after the bifurcations of chaos. We show power-law variation of maximal Lyapunov exponent near intermittency and sudden widening bifurcations. Linear variation is observed for band-merging bifurcation. We characterize weak and strong chaos using probability distribution of k-step dierence of a state variable.

In this paper, we present results of numerical experiments on chaotic transients in families of the logistic and Hénon maps. The duration of chaotic transients (the rambling time) for logistic maps estimated according to a rigorous criterion shows monotonic regularities with respect to both the period and the number of periodic window in a series of a given period. Due to inapplicability of this criterion to multidimensional maps, a more universal, though approximate, criterion is systematically studied on the family of logistic maps to optimize a choice of the free parameter value. The same approximate criterion is used to estimate rambling time for a number of periodic windows for the family of Hénon maps. The dependence of the rambling time on the width of periodic windows is tested.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1993

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.

2014

This paper serves as an introduction to the analysis of chaotic systems, with techniques being developed by working through two famous examples. The rst is the logistic map, a rst-order discrete dynamical system, and the second is the Lorenz system, a three-dimensional system of dierential equations.

Symmetry

In this work, a family of piecewise chaotic maps is proposed. This family of maps is parameterized by the nonlinear functions used for each piece of the mapping, which can be either symmetric or non-symmetric. Applying a constraint on the shape of each piece, the generated maps have no equilibria and can showcase chaotic behavior. This family thus belongs to the category of systems with hidden attractors. Numerous examples of chaotic maps are provided, showcasing fractal-like, symmetrical patterns at the interchange between chaotic and non-chaotic behavior. Moreover, the application of the proposed maps to a pseudorandom bit generator is successfully performed.