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<p>The phenomenon of drag reduction by polymers in turbulent flow has beenstudied over the last 60 years. New insight have been recently gained by meansof numerical simulation of dilute polymer solution at moderate values of theturbulent Reynolds number and elasticity. In this thesis, we track elastic parti-cles in Lagrangian frame in turbulent channel flow at Reτ = 180, by tracking,where the single particle obeys the FENE (finite extendible nonlinear elastic)formulation for dumbbel model. The feedback from polymers to the flow is notconsidered, while the Lagrangian approach enables us to consider high valuesof polymer elasticity. In addition, the finite time Lyapunov exponent (FTLE)of the flow is computed tracking infinitesimal material elements advected bythe flow. Following the large deviation theory, the Cramer’s function of theprobability density function of the FTLE for large values of time intervals isstudied at different wall-normal positions. The one-way effect of the turbulentflow on polymers is investigated by looking at the elongation and orientation ofthe polymers, with different relaxation times, across the channel. The confor-mation tensor of the polymers deformation which is an important contributionin the momentum balance equation is calculated by averaging in wall-parallelplanes and compared to theories available in the literature.</p> / QC 20100706

A complete method for calculating the largest Lyapunov exponent is developed in this thesis. For phase space reconstruction, a time delay estimator based on the average mutual information is discussed first. Then, embedding dimension is evaluated according to the False Nearest Neighbors algorithm. To obtain the parameters of all of the sub-functions and their derivatives, a multilayer feedforward neural network is applied to the time series data, after the time delay and embedding dimension are fixed. The Lyapunov exponents can be estimated using the Jacobian matrix and the QR decomposition. The possible applications of this method are then explored for various chaotic systems. Finally, the method is applied to some real world data to demonstrate the general relationship between the onset and progression of faults and changes in the largest Lyapunov exponent of a nonlinear system.

Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads. The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies. The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents. Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed. The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.

Many new materials used in mechanical and structural engineering exhibit viscoelastic properties, that is, stress depends on the past time history of strain, and vice versa. Investigating the behaviour of viscoelastic materials under dynamical loads is of great theoretical and practical importance for structural design, vibration reduction, and other engineering applications. The objective of this thesis is to find how viscoelasticity affects the stability of structures under random loads. The time history dependence of viscoelasticity renders the equations of motion of viscoelastic bodies in the form of integro-partial differential equations, which are more difficult to study compared to those of elastic bodies. The method of stochastic averaging, which has been proved to be an effective tool in the study of dynamical systems, is applied to simplify some single degree-of-freedom linear viscoelastic systems parametrically excited by wide-band noise and narrow-band noise. The solutions of the averaged systems are diffusion processes characterized by Itô differential equations. Therefore, the stability of the solutions is determined in the sense of the moment Lyapunov exponents and Lyapunov exponents, which characterize the moment stability and the almost-sure stability, respectively. The moment Lyapunov exponents may be obtained by solving the averaged Itô equations directly, or by solving the eigenvalue problems governing the moment Lyapunov exponents. Monte Carlo simulation is applied to study the behaviour of stochastic dynamical systems numerically. Estimating the moments of solutions through sample average may lead to erroneous results under the circumstances that systems exhibit large deviations. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented. Under certain conditions, the logarithm of norm of a solution converges weakly to normal distribution after suitably normalized. This property, along with the results of Komlós-Major-Tusnády for sums of independent random variables, are applied to construct the algorithm. The numerical results obtained from the improved algorithm are used to determine the accuracy of the approximate analytical moment Lyapunov exponents obtained from the averaged systems. In this way the effectiveness of the stochastic averaging method is confirmed. The world is essentially nonlinear. A single degree-of-freedom viscoelastic system with cubic nonlinearity under wide-band noise excitation is studied in this thesis. The approximated nonlinear stochastic system is obtained through the stochastic averaging method. Stability and bifurcation properties of the averaged system are verified by numerical simulation. The existence of nonlinearity makes the system stable in one of the two stationary states.

It is known that a highly ordered medium allows certain wave functions to move unhindered throughout and in this manner achieve delocalization. It is also known that if one introduces disorder into a medium, wave packets will not be able to move as freely and will instead be trapped or localized. In this thesis, I have simulated a medium in which the amount of disorder can be modified and using this I have shown that the shape of the localization can be altered.

In 2 ¡Ñ2 case,we discuss the uniqueness of the u-invariant measure on projective space.Under the condition that |detM|=1 for any M in Gu and Gu is not compact,we have the followings: (1) For any x in P(R^2),if #{M¡Dx|M belongs Gu}>2, then the u-invariant measure is unique. (2) For some x in P(R^2),there exists x1,x2 such that {M¡Dx|M belongs Gu} is contained in {x1,x2},if x1 and x2 are both fixed,then the u-invariant measure v is not unique;otherwise,if u has mass only on x1 and x2,then the u-invariant measure is unique.

Voltage flicker limitation of electric utilities has been discussed in the past three decades. Arc furnace is one of the most disturbing loads that cause flicker problems in the power network. If displeasing flicker levels are predictable, then corrective solution such as static var compensation or furnace controls could be developed in cooperation between the utility and the customer. In the past, the electric fluctuations in the arc furnace voltage have been proven to be chaotic in nature. This thesis proposes a phase space approach based on nonlinearity chaotic techniques to analyze and predict voltage flicker. The determination of the phase space dimension and the application of Lyapunov exponent for flicker prediction are described. Test results have shown that accurate prediction results are obtainable for short term flicker prediction based on chaos theory.

This dissertation thesis is dedicated to the heart rate variability and methods of its evaluation. It mainly focuses on nonlinear methods and especially on the Poincaré plot. First it deals with the principle and nature of the heart rate variability, then the ways of its representation, linear and also nonlinear methods of its analysis and physiological and pathophysiological influence on heart rate variability changes. In particular, there is emphasis on the metabolic syndrome. In the next section of the thesis there are compared and evaluated different ways of representation of the heart rate variability and further are tested selected methods of heart rate variability analysis on unique data from patients with the metabolic syndrome and healthy subjects provided by the Institute of Scientific Instruments, Academy of Sciences of Czech Republic. In particular, they are used the Poincaré plot and its parameters SD1 and SD2, commonly used time domain and frequency domain parameters, parameters evaluating signal entropy and the Lyapunov exponent. SD1 and SD2 combining the advantages of time and frequency domain methods of heart rate variability analysis distinguish successfully between patients with the metabolic syndrome and healthy subjects.