Probabilistic Method of Calculating the Maximum Lyapunov Exponent for Motion of Chaotic Systems

This paper introduces two types of Lorenz-like three-dimensional quadratic autonomous chaotic systems with 7 and 8 new parameters free of choice, respectively. Both systems are investigated at the equilibriums to study their chaotic characteristics. We focus our attention on the second type of the introduced system which consists of three nonlinear quadratic equations. Predictably, coordinates of the equilibriums are prohibitively complex. Therefore, instead of directly analyzing their stability, we prove the asymptotical characterization of equilibriums by utilizing our preliminary results derived for the first type of system. Our result shows that, though the coordinates of equilibriums satisfy a ternary quadratic, the system still contains only three equilibriums in circumstances of chaos. Sufficient conditions for the chaotic appearance of systems are derived. Our results are further verified by numerical simulations and the maximum Lyapunov exponent for several examples. Our research takes a first step in investigating chaos in Lorenz-like dynamic systems with strengthened nonlinearity and general forms of parameters.

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Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent

The Scientific World JOURNAL ◽

10.1155/2014/303614 ◽

2014 ◽

Vol 2014 ◽

pp. 1-16 ◽

Cited By ~ 2

Author(s):

Esteban Tlelo-Cuautle ◽

Ana Dalia Pano-Azucena ◽

Victor Hugo Carbajal-Gomez ◽

Mauro Sanchez-Sanchez

Keyword(s):

Phase Space ◽

Lyapunov Exponent ◽

Mathematical Description ◽

Chaotic Systems ◽

Chaotic Behavior ◽

Quantitative Measure ◽

Operational Amplifiers ◽

Chaotic Oscillator ◽

Maximum Lyapunov Exponent ◽

Experimental Realization

Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Hereina, b, c, d1are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.

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Non-Linearity and Chaotic Behaviour in Cyprus Stock Market

International Journal of Financial Management ◽

10.21863/ijfm/2015.5.4.018 ◽

2015 ◽

Vol 5 (4) ◽

Author(s):

Athina Bougioukou

Keyword(s):

Lyapunov Exponent ◽

Stock Market ◽

Chaos Theory ◽

Linear Dependence ◽

Chaotic Behavior ◽

Efficient Market Hypothesis ◽

Chaotic Behaviour ◽

Maximum Lyapunov Exponent ◽

Efficient Market ◽

General Index

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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Fatigue damage evaluation of metallic components based on the maximum Lyapunov exponent of modified Duffing system

Results in Physics ◽

10.1016/j.rinp.2021.104252 ◽

2021 ◽

pp. 104252

Author(s):

Yuhua Zhang ◽

Silong Quan ◽

Guoquan Liu

Keyword(s):

Lyapunov Exponent ◽

Fatigue Damage ◽

Maximum Lyapunov Exponent ◽

Damage Evaluation ◽

Duffing System

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Study on State Recognition of ASCE Benchmark Based on Lyapunov Exponent Spectrum Entropy

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.243-249.5435 ◽

2011 ◽

Vol 243-249 ◽

pp. 5435-5439 ◽

Cited By ~ 1

Author(s):

Jian Xi Yang ◽

Jian Ting Zhou ◽

Yue Chen

Keyword(s):

Lyapunov Exponent ◽

Time Sequence ◽

Maximum Lyapunov Exponent ◽

Entropy Analysis ◽

Structural System ◽

Time Duration ◽

State Recognition ◽

Chaotic Characteristic ◽

Lyapunov Exponent Spectrum ◽

Maximum Lyapunov Exponents

The paper has made a maximum Lyapunov exponent and Lyapunov exponent spectrum entropy analysis of ASCE Benchmark using non-linear theory and chaos time sequence. The maximum Lyapunov exponents in the two kinds of structural monitored data are both over zero, indicating that in the structural system chaos phenomenon has appeared. And, experiments have shown that the maximum Lyapunov exponent is sensitive of the amount of samples and the time delay. So, to compute the chaos index, the amount of samples and the time duration are of importance. Meanwhile, the Lyapunov exponent spectrum entropy is effective to measure the chaotic characteristic of the system, but ,the entropy is less sensitive to state recognition more than the max Lyapunov exponent.

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MUTUAL AND SELF-COMPOUNDING OF CHAOTIC SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496000424 ◽

1996 ◽

Vol 06 (04) ◽

pp. 759-767

Author(s):

R. SINGH ◽

P.S. MOHARIR ◽

V.M. MARU

Keyword(s):

Lyapunov Exponent ◽

Lyapunov Exponents ◽

Chaotic System ◽

Dynamic Systems ◽

Mantle Convection ◽

Chaotic Systems ◽

Compound System ◽

Two Systems

The notion of compounding a chaotic system was introduced earlier. It consisted of varying the parameters of the compoundee system in proportion to the variables of the compounder system, resulting in a compound system which has in general higher Lyapunov exponents. Here, the notion is extended to self-compounding of a system with a real-earth example, and mutual compounding of dynamic systems. In the former, the variables in a system perturb its parameters. In the latter, two systems affect the parameters of each other in proportion to their variables. Examples of systems in such compounding relationships are studied. The existence of self-compounding is indicated in the geodynamics of mantle convection. The effect of mutual compounding is studied in terms of Lyapunov exponent variations.

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