Strange attractor in earthquake swarms near Valsad (Gujarat), India

Valsad district in south Gujarat near the western coast of the peninsular India experienced earthquake swarms since early February 1986. Seismic monitoring through a network of micro earthquake seismographs showed a well concentrated seismic activity over an area of 7 × 10 km2 with the depth of foci extending from 1 to 15 km. A total number of 21,830 earthquakes were recorded during March 1986 to June 1988. The daily frequency of earthquakes for this period was utilized to examine deterministic chaos through evaluation of dimension of strange attractor and Lyapunov exponent. The low dimension of 2.1 for the strange attractor and positive value of the largest Lyapunov exponent suggest chaotic dynamics in Valsad earthquake swarms with at least 3 parameters for earthquake predictability. The results indicate differences in the characteristics of deterministic chaos in intraplate and interplate regions of India.

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Predictability of Indian Exchange Rates

The Journal of Prediction Markets ◽

10.5750/jpm.v12i3.1585 ◽

2019 ◽

Vol 12 (3) ◽

pp. 1-22

Author(s):

Radhika Prosad Datta ◽

Ranajoy Bhattacharyya

Keyword(s):

Lyapunov Exponent ◽

Exchange Rates ◽

Deterministic Chaos ◽

Time Frame ◽

Largest Lyapunov Exponent ◽

Major Spot ◽

New Information ◽

Out Of Sample ◽

Available Information ◽

Central Bank Interventions

In this paper we determine the extent of predictability of India’s major spot exchange rates by using the Lyapunov exponent. We first determine whether the series is fractal (self-similar) in nature. If it is indeed so, then next we determine whether the underlying dynamics of the system is deterministic or stochastic. If the dynamics is found to be deterministic then we calculate the Largest Lyapunov Exponent (LLE) to determine whether the series has deterministic chaos. Finally we use the inverse of the Lyapunov exponent to estimate the time period for which out of sample predictions for the series make sense. We find that India’s major spot exchange rates are: a) fractal in nature, b) chaotic with a high embedding dimension and c) The inverse of the LLE gives us a time frame in which any meaningful predictions can be made. These results are interpreted in two ways. First, exploiting the efficient market interpretation of randomness we conclude that since available information is fairly rapidly internalized, chaotic behaviour is mainly due to the unforeseen nature of the pool of new information affecting the systems at such short intervals of time. Second, anti-cyclical central bank interventions are conjectured to be the source of determinism in otherwise almost random movements.

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Chaotic dynamics in Parkfield region, California and characteristics earthquakes

MAUSAM ◽

10.54302/mausam.v50i1.1810 ◽

2021 ◽

Vol 50 (1) ◽

pp. 99-104

Author(s):

H. N. SRIVASTAVA ◽

K. C. SINHA RAY

Keyword(s):

Lyapunov Exponent ◽

Chaotic Dynamics ◽

Initial Conditions ◽

Strong Dependence ◽

Plate Boundary ◽

Deterministic Chaos ◽

Himalayan Region ◽

Positive Lyapunov Exponent ◽

Collision Type ◽

Earthquake Predictability

Based on about 75000 earthquakes in the California region detected through Parkfield network during the years 1969-1987, the occurrence of chaos was examined by two different approaches, namely, strange at tractor dimension and the Lyapunov exponent. The strange at tractor dimension was found as 6.3 in this region suggesting atleast 7 parameters for earthquake predictability. Small positive Lyapunov exponent of 0.045 provided further evidence for deterministic chaos in the region which showed strong dependence on the initial conditions. Implications of chaotic dynamics on characteristic Parkfield earthquakes has been discussed. The strange at tractor dimension in the region could be representative for the Transform type of plate boundary which is lower than that reported for continent collision type of plate boundary which is lower than that reported for continent collision type of plate boundary near Hindukush northwest Himalayan region.

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Largest Lyapunov Exponent and Attractors Applied to Stability Study of Nuclear Reactors

10.26678/abcm.cobem2017.cob17-0653 ◽

2017 ◽

Author(s):

WILMER ARUQUIPA COLOMA ◽

Antonella Lombardi Costa ◽

Claubia Pereira

Keyword(s):

Lyapunov Exponent ◽

Nuclear Reactors ◽

Stability Study ◽

Largest Lyapunov Exponent

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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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Study of Rehabilitation of Injured Knee Joint Applying Chaotic Theory in Human Body Motion

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.342-343.581 ◽

2007 ◽

Vol 342-343 ◽

pp. 581-584

Author(s):

Byung Young Moon ◽

Kwon Son ◽

Jung Hong Park

Keyword(s):

Time Series ◽

Lyapunov Exponent ◽

Angular Displacement ◽

Three Dimensional ◽

Gait Pattern ◽

Body Motion ◽

Largest Lyapunov Exponent ◽

Chaos Analysis ◽

Nonlinear Technique ◽

Injured Knee

Gait analysis is essential to identify accurate cause and knee condition from patients who display abnormal walking. Traditional linear tools can, however, mask the true structure of motor variability, since biomechanical data from a few strides during the gait have limitation to understanding the system. Therefore, it is necessary to propose a more precise dynamic method. The chaos analysis, a nonlinear technique, focuses on understanding how variations in the gait pattern change over time. Healthy eight subjects walked on a treadmill for 100 seconds at 60 Hz. Three dimensional walking kinematic data were obtained using two cameras and KWON3D motion analyzer. The largest Lyapunov exponent from the measured knee angular displacement time series was calculated to quantify local stability. This study quantified the variability present in time series generated from gait parameter via chaos analysis. Gait pattern is found to be chaotic. The proposed Lyapunov exponent can be used in rehabilitation and diagnosis of recoverable patients.

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A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416502266 ◽

2016 ◽

Vol 26 (13) ◽

pp. 1650226 ◽

Cited By ~ 20

Author(s):

Eduardo M. A. M. Mendes ◽

Erivelton G. Nepomuceno

Keyword(s):

Lyapunov Exponent ◽

Lower Bound ◽

Equations Of Motion ◽

Original System ◽

Glass System ◽

Largest Lyapunov Exponent ◽

Simple Method ◽

Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Hénon, Lorenz and Rössler equations and the Mackey–Glass system.

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