scholarly journals Dynamical properties of a novel one dimensional chaotic map

2022 ◽  
Vol 19 (3) ◽  
pp. 2489-2505
Author(s):  
Amit Kumar ◽  
◽  
Jehad Alzabut ◽  
Sudesh Kumari ◽  
Mamta Rani ◽  
...  
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Logistic Map ◽  
Maximal Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Dynamical Properties ◽  
Chaotic Logistic Map

<abstract><p>In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu &gt; 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.</p></abstract>

scholarly journals Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

2002 ◽  
Vol 7 (1) ◽  
pp. 41-52 ◽  
Author(s):  
A. M. López Jiménez ◽  
C. Camacho Martínez Vara de Rey ◽  
A. R. García Torres
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Direct Methods ◽  
Short Time Series ◽  
Direct Estimation ◽  
Non Linear ◽  
Short Time

The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation ofλstarting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.

New Discrete Chaotic Multiplicative Maps Based on the Logistic Map

2018 ◽  
Vol 28 (09) ◽  
pp. 1850118
Author(s):  
Dorota Aniszewska
Keyword(s):  
Mathematical Model ◽  
Chaotic Behavior ◽  
Nonlinear Models ◽  
Logistic Map ◽  
Point Of View ◽  
Bifurcation Diagrams ◽  
Deterministic Models ◽  
Multiplicative Calculus ◽  
Dynamical Properties ◽  
Stable Points

Chaos is a phenomenon which cannot be predicted if it manifests itself in a nonlinear system. Simple deterministic models, such as the logistic map [Formula: see text], are constructed to capture the essence of processes observed in nature. They are interesting also from a mathematical point of view: nonlinear models can behave in chaotic and complicated ways. The logistic map is the simplest mathematical model exhibiting chaotic behavior. Therefore, its dynamical properties, stable points and stable cycles are well known and widely described. In this paper, the properties of multiplicative calculus were employed to transform the classical logistic map into multiplicative ones. The multiplicative logistic maps were tested for chaotic behavior. The Lyapunov exponents together with the bifurcation diagrams are given.

COMPUTING THE DISTRIBUTION OF THE LYAPUNOV EXPONENT FROM TIME SERIES: THE ONE-DIMENSIONAL CASE STUDY

1995 ◽  
Vol 05 (06) ◽  
pp. 1721-1726 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Time Series ◽  
Monte Carlo ◽  
Lyapunov Exponent ◽  
Logistic Map ◽  
Observation Data ◽  
Block Bootstrap ◽  
One Dimensional ◽  
The One ◽  
True Values

In this paper, a simple and direct statistical method is proposed for estimating the Lyapunov exponent of an unknown dynamic system using its time series of observation data. It is shown that the asymptotic distribution of the estimates obtained from the proposed method is normal. Monte Carlo and block bootstrap methods are used to simulate the estimation for the logistic map, in which they both provide the expectation and variance for the estimates. Computer simulations show that our estimates are very close to the true values of the exponent for the logistic map with different parameters.

scholarly journals Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 474 ◽  
Author(s):  
Lazaros Moysis ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jesus M. Munoz-Pacheco ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Image Encryption ◽  
Fuzzy Numbers ◽  
Statistical Tests ◽  
Logistic Map ◽  
Simple Rule ◽  
Pseudorandom Number ◽  
Bifurcation Diagrams ◽  
Triangular Numbers ◽  
Pseudorandom Number Generation

A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption.

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

scholarly journals A Chaos based image Encryption Scheme using one Dimensional Exponential Logistic Map.

2019 ◽  
Vol 1 ◽  
pp. 223-237
Author(s):  
Terlumun Gbaden
Keyword(s):  
Correlation Coefficient ◽  
Chaotic Map ◽  
Logistic Map ◽  
Encryption Scheme ◽  
One Dimensional ◽  
Encryption And Decryption ◽  
Secret Keys ◽  
Encryption Decryption ◽  
Logistic Chaotic Map ◽  
Encrypted Images

The widespread use of images in various sectors of life makes its protection increasingly necessary and important. An improvement over encryption and decryption algorithm using exponential logistic chaotic map was proposed. In this work, we adopt an encryption/decryption strategy for colour images using the exponential logistic chaotic map. The proposed encryption/decryption algorithms are implemented in MATLAB for computer simulation. The experimental results indicate that the proposed algorithms can be used successfully to encrypt/decrypt images with secret keys. The performance analysis using histogram uniformity analysis and correlation coefficient show that the algorithms give larger space, quick speed and easy to realize. The encrypted images have good encryption effect and low correlation coefficient rendering it a good candidate for confidential and secure means of transmitting image information in untrusted networks.

A Chaotification model based on sine and cosecant functions for enhancing chaos

2021 ◽  
Author(s):  
Yuqing Li ◽  
Xing He ◽  
Dawen Xia
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Chaotic Maps ◽  
Dynamic Properties ◽  
Chaotic Map ◽  
Sample Entropy ◽  
One Dimensional ◽  
Model Based ◽  
Chaotic Region ◽  
Definition Of

Chaotic maps with higher chaotic complexity are urgently needed in many application scenarios. This paper proposes a chaotification model based on sine and cosecant functions (CMSC) to improve the dynamic properties of existing chaotic maps. CMSC can generate a new map with higher chaotic complexity by using the existing one-dimensional (1D) chaotic map as a seed map. To discuss the performance of CMSC, the chaos properties of CMSC are analyzed based on the mathematical definition of the Lyapunov exponent (LE). Then, three new maps are generated by applying three classical 1D chaotic maps to CMSC respectively, and the dynamic behaviors of the new maps are analyzed in terms of fixed point, bifurcation diagram, sample entropy (SE), etc. The results of the analysis demonstrate that the new maps have a larger chaotic region and excellent chaotic characteristics.

A cryptosystem for lossless/lossy grayscale images in IWT domain using chaotic map-based generated key matrices

2018 ◽  
Vol 16 (04) ◽  
pp. 1850024 ◽  
Author(s):  
Kshiramani Naik ◽  
Arup Kumar Pal
Keyword(s):  
Chaotic Map ◽  
Logistic Map ◽  
Security Analysis ◽  
Image Data ◽  
Natural Images ◽  
Security Level ◽  
Original Image ◽  
Data Set ◽  
The Individual ◽  
Chaotic Logistic Map

In this paper, an image encryption scheme based on reversible integer wavelet transform (IWT) with chaotic logistic map is designed. The proposed cryptosystem is applicable to encipher both the medical and natural images in lossless and lossy manners, respectively. Initially, the original image is transformed with the multilevel of IWT, then the image data set is divided into low sub-band (approximation part) and high sub-bands (detail part). The approximation part gets confused with the chaotic logistic map followed by the bit plane decomposition. Next, the individual bit planes are further diffused with several binary key metrics, generated using a chaotic logistic map. The proposed key schedule derives several large size of binary key metrics from a small size of key. Based on the type of applications, the detail part is considered for lossless/lossy compression. The lossless/lossy compressed detail part is further considered only for confusion process using the logistic map for the sake of enhancing the security level. Finally, the cipher image obtained after inverse IWT is significantly dissimilar than original image. The scheme has been tested on several standard medical and natural images and the experimental results substantiate that a small size of key is enough to protect the content of images completely. The security analysis reveals that the proposed scheme is suitable for protecting the image data effectively.

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