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. / QC 20100706

Nesta dissertação estudamos de um ponto de vista probabilístico, o comportamento assintótico de sistemas dinâmicos. Um exemplo simples de formular e profundo é o estudo de produto aleatório de matrizes (FURSTENBERG; KESTEN, 1960). Utilizaremos como ferramenta o estudo dos cociclos lineares, posteriormente mediante o Teorema de Furstenberg-Kesten definiremos o expoente de Lyapunov do cociclo, em seguida enunciamos e provamos o Teorema Ergódico Multiplicativo de Oseledets o qual nos permite entender o comportamento das órbitas típicas para um cociclo dado F : M x R2 → M x R2. O Teorema de Fusrtenberg-Kesten fornece informações sobre o crescimento das matrizes An(x), enquanto o Teorema de Oseledets descreve o comportamento assintótico dos vetores An(x).v. Finalmente provamos o teorema principal desta dissertação, o Teorema de Furstenberg o qual diz que na maioria dos casos o maior expoente de Lyapunov é positivo (FURSTENBERG, 1963). / In this thesis we study from a probabilistic point of view, the asymptotic behavior of dynamic systems, a deep and simple example is the random product of matrices (FURSTENBERG; KESTEN, 1960). We will use as a tool, the study of linear cocycles, later using the Furstenberg- Kesten Theorem we will define the Lyapunov exponent of the cocycle, then we enunciate and prove the Multiplicative Ergodic Theorem of Oseledets which allow us to understand the behavior of the typical orbits for a given cocycle F : M x R2 → M x R2. The Fusrtenberg- Kesten theorem provides information on the growth of the matrices A(x), while the theorems of Oseledets describe the asymptotic behavior of the vectors An(x).v. Finally we prove our main theorem, Furstenbergs Theorem which states that in most cases the greatest exponent of Lyapunov is positive (FURSTENBERG, 1963).

Nesta dissertação estudamos de um ponto de vista probabilístico, o comportamento assintótico de sistemas dinâmicos. Um exemplo simples de formular e profundo é o estudo de produto aleatório de matrizes (FURSTENBERG; KESTEN, 1960). Utilizaremos como ferramenta o estudo dos cociclos lineares, posteriormente mediante o Teorema de Furstenberg-Kesten definiremos o expoente de Lyapunov do cociclo, em seguida enunciamos e provamos o Teorema Ergódico Multiplicativo de Oseledets o qual nos permite entender o comportamento das órbitas típicas para um cociclo dado F : M x R2 → M x R2. O Teorema de Fusrtenberg-Kesten fornece informações sobre o crescimento das matrizes An(x), enquanto o Teorema de Oseledets descreve o comportamento assintótico dos vetores An(x).v. Finalmente provamos o teorema principal desta dissertação, o Teorema de Furstenberg o qual diz que na maioria dos casos o maior expoente de Lyapunov é positivo (FURSTENBERG, 1963). / In this thesis we study from a probabilistic point of view, the asymptotic behavior of dynamic systems, a deep and simple example is the random product of matrices (FURSTENBERG; KESTEN, 1960). We will use as a tool, the study of linear cocycles, later using the Furstenberg- Kesten Theorem we will define the Lyapunov exponent of the cocycle, then we enunciate and prove the Multiplicative Ergodic Theorem of Oseledets which allow us to understand the behavior of the typical orbits for a given cocycle F : M x R2 → M x R2. The Fusrtenberg- Kesten theorem provides information on the growth of the matrices A(x), while the theorems of Oseledets describe the asymptotic behavior of the vectors An(x).v. Finally we prove our main theorem, Furstenbergs Theorem which states that in most cases the greatest exponent of Lyapunov is positive (FURSTENBERG, 1963).

<p>The numerical path integration method for solving stochastic differential equations is extended to solve systems up to six spatial dimensions, angular variables, and highly nonlinear systems - including systems that results in discontinuities in the response probability density function of the system. Novel methods to stabilize the numerical method and increase computation speed are presented and discussed. This includes the use of the fast Fourier transform (FFT) and some new spline interpolation methods. Some sufficient criteria for the path integration theory to be applicable is also presented. The development of complex numerical code is made possible through automatic code generation by scripting. The resulting code is applied to chaotic dynamical systems by adding a Gaussian noise term to the deterministic equation. Various methods and approximations to compute the largest Lyapunov exponent of these systems are presented and illustrated, and the results are compared. Finally, it is shown that the location and size of the additive noise term affects the results, and it is shown that additive noise for specific systems could make a non-chaotic system chaotic, and a chaotic system non-chaotic.</p>

The numerical path integration method for solving stochastic differential equations is extended to solve systems up to six spatial dimensions, angular variables, and highly nonlinear systems - including systems that results in discontinuities in the response probability density function of the system. Novel methods to stabilize the numerical method and increase computation speed are presented and discussed. This includes the use of the fast Fourier transform (FFT) and some new spline interpolation methods. Some sufficient criteria for the path integration theory to be applicable is also presented. The development of complex numerical code is made possible through automatic code generation by scripting. The resulting code is applied to chaotic dynamical systems by adding a Gaussian noise term to the deterministic equation. Various methods and approximations to compute the largest Lyapunov exponent of these systems are presented and illustrated, and the results are compared. Finally, it is shown that the location and size of the additive noise term affects the results, and it is shown that additive noise for specific systems could make a non-chaotic system chaotic, and a chaotic system non-chaotic.

An analog controller has been analyzed and built for an electrostatic micro-cantilever beam. The closed loop MEMS device can be used as both actuator and sensor. As an actuator it will have the advantage of large stable travel range up to 90% of the gap. As a sensor the beam is to be driven into chaotic motion which is very sensitive changes in the system parameters. Two versions of the controller have been analyzed and implemented, one for the actuator and one for the sensor. For the actuator, preliminary experiments show good matching with the model. As for the sensor, the dynamic behavior have been studied and the best operating regions have been determined.

Voltage (lighting) flicker is mainly caused by the electric arc furnaces (EAF) facility supplied by the medium and high voltage power network. In addition to that, because of the increase of wind power generation in both quantity and capacity, intermittent power output of wind turbines under wind speed variation could also cause voltage flickers that affect the performance of lighting and electronics devices in the neighboring feeder buses. Successful voltage flicker prediction and propagation estimation would help both utility and customers in dealing with the problem. This dissertation presents a nonlinear model for the short term prediction of voltage flicker due to EAF operations. In this study, synchronized voltage flicker measurement was conducted at several EAF facilities to understand the stochastic behavior of voltage flicker. The electric loading condition during EAF melting process shows a long term qualitative behavior of a dynamic system and illustrates a special structure of a fractal system. With the fractal structure identification, the behavior hidden behind the voltage flicker time series measurement could be grasped. Using a phase space reconstruction technique and Lyapunov exponent (LE) of state trajectory in the phase space, based on actual voltage flicker measurements, it is proved that the voltage flicker time series is chaos. By using LE, three formulations are adopted to build the prediction models and illustrate the feasibility of short term EAF voltage flicker prediction. Currently, some Asian countries are using the Japanese £GV10 flicker voltage standard. Due to the adoption of IEC standard by IEEE and European countries, a rational conversion of flicker planning limits between different standards would help utilities consider revising or changing their voltage flicker standards and planning limits. Statistical analyses of Pst and £GV10 measurement are conducted in this study. Under different EAF types and operation conditions, reasonable conversion factors between Pst and £GV10 standards are derived, and the flicker transfer factor between different voltage levels of the power supply system are presented.

Periodic and chaotic dynamical systems follow deterministic equations such as Newton's laws of motion. To distinguish the difference between two systems, the initial conditions have an important role. Chaotic behaviors or dynamics are characterized by sensitivity to initial conditions. Mathematically, a chaotic system is defined as a system very sensitive to initial conditions. A small difference in initial conditions causes unpredictability in the final outcome. If error is measured from the initial state, the relative error grows exponentially. Prediction becomes impossible and finally, chaotic systems can come to become stochastic system. To make chaotic motion, the number of variables in the system should be above three and there should be non-linear terms coupling several of the variables in the equation of motion. Phase space is defined as the space spanned by the coordinate and velocity vectors. In our case, mixing zone is phase space. With the above characteristics - the initial condition sensitivity of a chaotic system, our plan is to find most efficient chaotic stirrer. In this thesis, we present four methods to measure mixing state based on the chaotic dynamics theory. The Lyapunov exponent is a measure of the sensitivity to initial conditions and can be used to calculate chaotic strength. We can decide the chaotic state with one real number and measure efficiency of the chaotic mixer and find the optimum frequency. The Poincare section method provides a means for viewing the phase space diagram so that the motion is observed periodically. To do this, the trajectory is sectioned at regular intervals. With the Poincare section method, we can find 'islands' considered as bad mixed zones so that the mixing state can be measured qualitatively. With the chaotic dynamics theory, the initial length of the interface can grow exponentially in a chaotic system. We will show the above characteristics of the chaotic system to prove as fact that our model is an efficient chaotic mixer. The final goal for making chaotic stirrer is how to implement efficient dispersed particles. The box counting method is focused on measurement of the particles dispersing state. We use snap shots of the mixing process and with these snap shots, we devise a plan to measure particles' dispersing rate using the box-counting method.

The first part of this work gives an introduction into aperiodic order in general and the lines of research pursued. The second part consists of eight manuscripts.

An analog controller has been analyzed and built for an electrostatic micro-cantilever beam. The closed loop MEMS device can be used as both actuator and sensor. As an actuator it will have the advantage of large stable travel range up to 90% of the gap. As a sensor the beam is to be driven into chaotic motion which is very sensitive changes in the system parameters. Two versions of the controller have been analyzed and implemented, one for the actuator and one for the sensor. For the actuator, preliminary experiments show good matching with the model. As for the sensor, the dynamic behavior have been studied and the best operating regions have been determined.