scholarly journals An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1511
Author(s):  
Kei Inoue
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Chaotic Maps ◽  
Chaotic Map ◽  
Calculation Formula ◽  
Two Dimensional ◽  
Information Dynamics ◽  
The Difference ◽  
Chaos Degree

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.

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.

Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950030 ◽  
Author(s):  
Fahimeh Nazarimehr ◽  
Aboozar Ghaffari ◽  
Sajad Jafari ◽  
Seyed Mohammad Reza Hashemi Golpayegani
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dictionary Learning ◽  
Nonlinear Dynamical Systems ◽  
Sparse Recovery ◽  
Governing Equations ◽  
Nonlinear Dynamical ◽  
Bifurcation Points ◽  
Long Time

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

scholarly journals Quantifiers for randomness of chaotic pseudo-random number generators

2009 ◽  
Vol 367 (1901) ◽  
pp. 3281-3296 ◽  
Author(s):  
L. De Micco ◽  
H. A. Larrondo ◽  
A. Plastino ◽  
O. A. Rosso
Keyword(s):  
Time Series ◽  
Comparative Analysis ◽  
Invariant Measure ◽  
Random Number ◽  
Chaotic Maps ◽  
Chaotic Map ◽  
General Purpose ◽  
Statistical Characteristics ◽  
Random Number Generators ◽  
Pseudo Random Number

We deal with randomness quantifiers and concentrate on their ability to discern the hallmark of chaos in time series used in connection with pseudo-random number generators (PRNGs). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely (i) its invariant measure and (ii) the mixing constant. This is of help in answering two questions that arise in applications: (i) which is the best PRNG among the available ones? and (ii) if a given PRNG turns out not to be good enough and a randomization procedure must still be applied to it, which is the best applicable randomization procedure? Our answer provides a comparative analysis of several quantifiers advanced in the extant literature.

scholarly journals An extension of the entropic chaos degree and its positive effect

2021 ◽  
Author(s):  
Kei Inoue ◽  
Tomoyuki Mao ◽  
Hidetoshi Okutomi ◽  
Ken Umeno
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponent ◽  
Initial Point ◽  
Estimation Methods ◽  
Dimensional Euclidean Space ◽  
Information Dynamics ◽  
Finite Space ◽  
Definition Of ◽  
Positive Effect ◽  
Chaos Degree

AbstractThe Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing the exponential sensitivity of an initial point on the dynamical system. However, we cannot directly compute the Lyapunov exponent for a dynamical system without its dynamical equation, although some estimation methods do exist. Information dynamics introduces the entropic chaos degree to measure the strength of chaos of the dynamical system. The entropic chaos degree can be used to compute the strength of chaos with a practical time series. It may seem like a kind of finite space Kolmogorov-Sinai entropy, which then indicates the relation between the entropic chaos degree and the Lyapunov exponent. In this paper, we attempt to extend the definition of the entropic chaos degree on a d-dimensional Euclidean space to improve the ability to measure the stength of chaos of the dynamical system and show several relations between the extended entropic chaos degree and the Lyapunov exponent.

scholarly journals Investigation of the difference between Chaos Degree and Lyapunov exponent for asymmetric tent maps

JSIAM Letters ◽  
2019 ◽  
Vol 11 (0) ◽  
pp. 61-64 ◽  
Author(s):  
Tomoyuki Mao ◽  
Hidetoshi Okutomi ◽  
Ken Umeno
Keyword(s):  
Lyapunov Exponent ◽  
Tent Maps ◽  
The Difference ◽  
Chaos Degree

Analytical Solutions to Multivalued Maps

1997 ◽  
Vol 11 (12) ◽  
pp. 521-530 ◽  
Author(s):  
Jorge A. Gonzalez ◽  
Lindomar B. De Carvalho
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Analytical Solutions ◽  
Chaotic Maps ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Maximum Lyapunov Exponent ◽  
Multivalued Maps ◽  
Return Maps ◽  
Set Of Points

We present explicit solutions for a class of chaotic maps. The return-maps generated by a special class of chaotic functions can be multivalued, or even they can represent an erratic set of points. In some cases the produced time series can have an increasing time-dependent maximum Lyapunov exponent. We discuss some applications of the obtained results. In particular, we present a chaotic lattice model for the investigation of the propagation of carriers in the presence of disorder.

Fractional two-dimensional discrete chaotic map and its applications to the information security with elliptic-curve public key cryptography

2017 ◽  
Vol 24 (20) ◽  
pp. 4797-4824 ◽  
Author(s):  
Zeyu Liu ◽  
Tiecheng Xia ◽  
Jinbo Wang
Keyword(s):  
Elliptic Curve ◽  
Color Image ◽  
Chaotic Map ◽  
Public Key Cryptography ◽  
Public Key ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Two Dimensional ◽  
Function Combination ◽  
The Difference

A novel fractional two-dimensional triangle function combination discrete chaotic map is proposed by use of the discrete fractional calculus. The chaos behaviors are then discussed when the difference order is a fractional one. The bifurcation diagrams, the largest Lyapunov exponent and the phase portraits are displayed, especially, the elliptic curve public key cryptosystem is used in color image encryption algorithm.

scholarly journals A RETURN MAP REGRESSION APPROACH FOR NONCOHERENT DETECTION IN CHAOTIC DIGITAL COMMUNICATIONS

2003 ◽  
Vol 13 (03) ◽  
pp. 685-690 ◽  
Author(s):  
CHI K. TSE ◽  
FRANCIS C. M. LAU
Keyword(s):  
Good Accuracy ◽  
Chaotic Maps ◽  
Additive White Gaussian Noise ◽  
Chaotic Map ◽  
Noncoherent Detection ◽  
Return Maps ◽  
Regression Approach ◽  
Shift Keying ◽  
The Difference ◽  
Chaos Shift Keying

Chaos-based communications can be applied advantageously if the property of chaotic systems is suitably exploited. In this Letter a simple noncoherent detection method for chaos-shift-keying (CSK) modulation is proposed, exploiting some distinguishable property of chaotic maps for recovering the digital message. Specifically, the proposed method exploits the difference in the return maps of the signals representing the digital symbols. The determining parameter of the return maps is estimated using a simple regression algorithm. If the parameter strongly characterizes the chaotic map, the detection can achieve very good accuracy. This strong parametric characterization can be achieved by defining the regression model with only one parameter. Using the tent maps as chaos generators, the bit-error-rate under additive white Gaussian noise is studied by computer simulations.

scholarly journals A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 103
Author(s):  
Bulcsú Sándor ◽  
Bence Schneider ◽  
Zsolt I. Lázár ◽  
Mária Ercsey-Ravasz
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
State Transition ◽  
Transition Probabilities ◽  
Coarse Graining ◽  
Special Focus ◽  
Entropy Of Transition ◽  
Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

scholarly journals An Algorithm to Automatically Detect the Smale Horseshoes

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Qingdu Li ◽  
Lina Zhang ◽  
Fangyan Yang
Keyword(s):  
Dynamical Systems ◽  
Chaotic Maps ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Unstable Manifolds ◽  
Duffing Map

Smale horseshoes, curvilinear rectangles and their U-shaped images patterned on Smale's famous example, provide a rigorous way to study chaos in dynamical systems. The paper is devoted to constructing them in two-dimensional diffeomorphisms with the existence of transversal homoclinic saddles. We first propose an algorithm to automatically construct “horizontal” and “vertical” sides of the curvilinear rectangle near to segments of the stable and of the unstable manifolds, respectively, and then apply it to four classical chaotic maps (the Duffing map, the Hénon map, the Ikeda map, and the Lozi map) to verify its effectiveness.

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