Using the Lyapunov Exponent as a Practical Test for Noisy Chaos

The intention of this research is to investigate the aspect of non-linearity and chaotic behavior of the Cyprus stock market. For this purpose, we use non-linearity and chaos theory. We perform BDS, Hinich-Bispectral tests and compute Lyapunov exponent of the Cyprus General index. The results show that existence of non-linear dependence and chaotic features as the maximum Lyapunov exponent was found to be positive. This study is important because chaos and efficient market hypothesis are mutually exclusive aspects. The efficient market hypothesis which requires returns to be independent and identically distributed (i.i.d.) cannot be accepted.

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

Entropy ◽

10.3390/e22040474 ◽

2020 ◽

Vol 22 (4) ◽

pp. 474 ◽

Cited By ~ 5

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022640 ◽

2008 ◽

Vol 18 (12) ◽

pp. 3679-3687 ◽

Cited By ~ 6

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.

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The Lyapunov Exponent Variance of an Electronic Manufacturing Enterprise’s Daily Qualified Rate Time Series by Improved Small Data Sets Approach

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.197.271 ◽

2012 ◽

Vol 197 ◽

pp. 271-277

Author(s):

Zhu Ping Gong

Keyword(s):

Time Series ◽

Lyapunov Exponent ◽

Chaotic Time Series ◽

Quality System ◽

Quality Level ◽

Largest Lyapunov Exponent ◽

Small Data ◽

Data Set ◽

Electronic Manufacturing ◽

Small Data Set

Small data set approach is used for the estimation of Largest Lyapunov Exponent (LLE). Primarily, the mean period drawback of Small data set was corrected. On this base, the LLEs of daily qualified rate time series of HZ, an electronic manufacturing enterprise, were estimated and all positive LLEs were taken which indicate that this time series is a chaotic time series and the corresponding produce process is a chaotic process. The variance of the LLEs revealed the struggle between the divergence nature of quality system and quality control effort. LLEs showed sharp increase in getting worse quality level coincide with the company shutdown. HZ’s daily qualified rate, a chaotic time series, shows us the predictable nature of quality system in a short-run.

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