Macro-instability: a chaotic flow component in stirred tanks

2007 ◽  
Vol 366 (1864) ◽  
pp. 409-418 ◽  
Author(s):  
Pavel Hasal ◽  
Milan Jahoda ◽  
Ivan Fořt
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Fluid Velocity ◽  
Tangential Force ◽  
Axial Component ◽  
Maximum Lyapunov Exponent ◽  
Stirred Tanks ◽  
Operational Conditions ◽  
Rushton Turbine ◽  
Proper Orthogonal

Chaotic features of the macro-instability (MI) of flow patterns in stirred tanks are studied in this paper. Datasets obtained by measuring the axial component of the fluid velocity and the tangential force affecting the baffles are used. Two geometrically identical, flat-bottomed cylindrical mixing tanks (diameter of 0.3 m) stirred with either pitched blade turbine impellers or Rushton turbine impeller are used in the experiments, and water and aqueous glycerol solutions are used as the working liquids. First, the presence of the MI component in the data is examined by spectral analysis. Then, the MI components are identified in the data using the proper orthogonal decomposition (POD) technique. The attractors of the macro-instability are reconstructed using either the POD eigenmodes or a method of delays and finally the attractor invariants are evaluated. The dependence of the correlation dimension and maximum Lyapunov exponent on the vessel operational conditions is determined together with their distribution within the tank. No significant spatial variability of the correlation dimension value is observed. Its value is strongly influenced by impeller speed and by the vessel–impeller geometry. More profound spatial distribution is displayed by the maximum Lyapunov exponent taking distinctly positive values. These two invariants, therefore, can be used to locate distinctive regions with qualitatively different MI dynamics within the stirred tank.

Non-Linearity and Chaotic Behaviour in Cyprus Stock Market

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.

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.

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

2011 ◽  
Vol 243-249 ◽  
pp. 5435-5439 ◽  
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.

scholarly journals The Tangential Force Affecting the Radial Baffles in a Stirred Vessel: Analysis of the Macro-instability Related Component

10.14311/450 ◽  
2003 ◽  
Vol 43 (4) ◽  
Author(s):  
P. Hasal ◽  
I. Fořt ◽  
J. Kratěna
Keyword(s):  
Experimental Data ◽  
Spectral Analysis ◽  
Tangential Force ◽  
Low Frequency ◽  
Frequency Component ◽  
Relative Magnitude ◽  
Tangential Component ◽  
Vertical Position ◽  
Total Force ◽  
Rushton Turbine

Experimental data obtained by measuring the tangential component of the force affecting radial baffles in a flat-bottomed cylindrical mixing vessel stirred with a Rushton turbine impeller is analysed. Spectral analysis of the experimental data demonstrated the presence of its macro-instability (MI) related low-frequency component embedded in the total force. Two distinct dimensionless frequencies (both directly proportional to the impeller speed of rotation N) of the occurence of the MI component were detected: a lower frequency of approximately 0.025N and a higher frequency of about 0.085N. The relative magnitude QMI of the MI-related component of the total tangential force was evaluated by a combination of proper orthogonal decomposition (POD) and spectral analysis. The values of magnitude QMI varied in the interval [rom approximately 0.05 to 0.30. The magnitude QMI takes maximum values at low Reynolds number values (in laminar and transitional regions). In the turbulent region (ReM >20000) the QMI value is low and practically constant. The dependence oj the QMI values on vertical position in the vessel is only marginal. The results suggest that the magnitude of the MI component of the force is significantly influenced by the liquid viscosity and density.

scholarly journals An investigation of EEG dynamics in an animal model of temporal lobe epilepsy using the maximum Lyapunov exponent

2009 ◽  
Vol 216 (1) ◽  
pp. 115-121 ◽  
Author(s):  
Sandeep P. Nair ◽  
Deng-Shan Shiau ◽  
Jose C. Principe ◽  
Leonidas D. Iasemidis ◽  
Panos M. Pardalos ◽  
...  
Keyword(s):  
Animal Model ◽  
Lyapunov Exponent ◽  
Temporal Lobe Epilepsy ◽  
Temporal Lobe ◽  
Maximum Lyapunov Exponent

Chaotic Dynamics of Piezoelectric MEMS Based on Maximum Lyapunov Exponent and Smaller Alignment Index Computations

2020 ◽  
Vol 30 (09) ◽  
pp. 2030025
Author(s):  
M. V. Tchakui ◽  
P. Woafo ◽  
Ch. Skokos
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic Dynamics ◽  
Detection Method ◽  
Dynamical Behavior ◽  
Time Dependent ◽  
Maximum Lyapunov Exponent ◽  
Nonconservative System ◽  
Surface Of Section ◽  
Chaos Detection ◽  
Piezoelectric Mems

We characterize the dynamical states of a piezoelectric micrcoelectromechanical system (MEMS) using several numerical quantifiers including the maximum Lyapunov exponent, the Poincaré Surface of Section and a chaos detection method called the Smaller Alignment Index (SALI). The analysis makes use of the MEMS Hamiltonian. We start our study by considering the case of a conservative piezoelectric MEMS model and describe the behavior of some representative phase space orbits of the system. We show that the dynamics of the piezoelectric MEMS becomes considerably more complex as the natural frequency of the system’s mechanical part decreases. This refers to the reduction of the stiffness of the piezoelectric transducer. Then, taking into account the effects of damping and time-dependent forces on the piezoelectric MEMS, we derive the corresponding nonautonomous Hamiltonian and investigate its dynamical behavior. We find that the nonconservative system exhibits a rich dynamics, which is strongly influenced by the values of the parameters that govern the piezoelectric MEMS energy gain and loss. Our results provide further evidences of the ability of the SALI to efficiently characterize the chaoticity of dynamical systems.

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