Efficient mixing can be achieved in flows where turbulence is absent, if the trajectories of passively advected particles in the flow are chaotic. The chaotic nature of particle trajectories results in exponential stretching of material lines in the flow. Thus the interface along which diffusion occurs is stretched exponentially leading to efficient mixing. It has been demonstrated recently that regions in flow fields that exhibit poor mixing and non-chaotic particle trajectories can have an important bearing on the overall dynamics and transport of the entire domain. The space-time trajectories of physical stirrers or elliptic points in two dimensional flows can be classified according to braid groups. One can predict a lower bound on the topological entropy (i.e. exponential rate of stretching of material lines) of flows (h_f) by applying the Thurston-Nielsen classification theorems to these braids. This gives a reduced order model for the dynamics of transport of the entire flow field using just a few points. Recent work has shown that this methodology can be used to estimate a lower bound on h_f using the braids formed by Almost Cyclic Sets (ACS) in certain periodic Stokes' flows. These ACS are closely related to Almost Invariant Sets (AIS) which are identified using a probabilistic set oriented method that makes use of the descritised Perron-Frobenius operator of the flow map. This work extends this approach to flows at non-zero Reynolds numbers, which take into account the effects of inertia. The role of Finite Time Coherent Structures (FTCS) in the dynamics of flow fields is also investigated. Unlike ACS, the FTCS approach is more general as it can be applied to aperiodic flow fields. Further, the relationship between mixing efficiency and the topological entropy of flow fields at non-zero Reynolds numbers is also studied. / Ph. D.

Four aspects of the dynamics of continuous-time dynamical systems are studied in this work. The relationship between the Lyapunov exponents of the original system and the Lyapunov exponents of induced Poincare maps is examined. The behavior of these Poincare maps as discriminators of chaos from noise is explored, and the possible Poissonian statistics generated at rarely visited surfaces are studied.

This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts. In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable. The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

published_or_final_version / Business Administration / Master / Master of Business Administration

No description available.

This discussion of chaos theory is concerned with two major issues. On the one hand, I explore what kind of knowledge is linked to chaos theory, and more specifically how as a science it informs the cultural discourses created by postindustrial societies. On the other hand, I probe chaos theory's potential as a model for challenging the existing conception of our world within the prevailing epistemologies of order and predictability. Both of these issues are addressed with in mind the broader framework and question concerning social relations, especially to the extent that those relations, in their spatial dimension, have become an object of scientific discourse. My approach to chaos theory is purposefully eclectic, conjoining the scientific with the social and the political. I believe that chaos theory points to a dynamic, intertextual, and multidimensional universe, and therefore, my interest lies in these connections, in bridging the various elements working together to create our contemporary, postmodern world. Science creates theories and images of nature that have been used to subordinate and control segments of the population through theories of race and sexuality. Thus, to recognize the existence of complexity and instability is to give away powerful conceptual means of political and social control, a strategy in which Western science has been an active participant.

We are interested in examining the long-time decay rate of a passive scalar in two-dimensional flows. The focus is on the effect of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary C of the domain D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar, for example temperature, is exponential in time with a decay rate gamma(kappa), where kappa is the molecular diffusivity. Scaling laws of the form gamma(kappa) ~ C*kappa^alpha for small kappa are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, alpha ~ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that alpha ~ 1/2 for a plane layer, but find alpha ~ 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is alpha ~ 1/2 for a slip flow condition and alpha ~ 3/4 for a no-slip condition. The scaling law exponents alpha for chaotic time-periodic flows are compared with those for similarly constructed random flows. Motivated by the theory of passive scalar field, in Part II of this work we extend the investigation of the evolution of passive scalar for the flows addressed specifically in Part I. Based on an ensemble averaging over random velocity fields, the theoretical results obtained confirm the scaling laws computed numerically for a single, long realisation of random flows. In analogy with Lebedev and Turitsyn (2004) and Salman and Haynes (2007) our results show very good agreement between such an ensemble theory and applications. In part III of our study, we expand upon the work set out in the previous parts of this thesis in terms of the polar-co-ordinate system. We analyse the structures of flows driven near to a corner with a link to Moffatt corner eddies. A long-time exponential decay rate gamma(kappa)=C*kappa^alpha has been obtained confirming our numerical and theoretical results predicted in Part I and Part II in this work. The exponent alpha is determined in a structure of Moffatt corner eddies.

ΛCDM cosmology is described in terms of general relativity and the Robertson-Walker metric. The evolution of the observable universe, currently dominated by dark energy (Λ) and cold dark matter (CDM), is presented in terms of its thermal history. CDM is extended to include an inflation epoch that accelerates the early expansion rate to near exponential levels. It is shown that inflation solves several problems in CDM and produces perturbations in the metric that lead to the observed anisotropies in the Cosmic Microwave Background and the formation of large scale cosmological structures. Various theories of inflation are explored. Predictions of inflation theories are compared to observations published by the Planck Collaboration. The paper concludes with an examination of “𝜶-attractor” theories of inflation based on a modified form of gravity.

Artist Statement My art is about seeking answers to personal conflicts while telling a story of a chaotic journey. I reflect on everyday moments and my thoughts as I discover ways to make sense of situations and life. I do this by creating textural, vibrantly colored, and gestural surfaces that emulate the powerful waters of the seas. I want to reveal an emotionally driven and process-oriented experience to the viewer. While creating, I do not maintain full control of the media and let the process become part of the work. I aggressively layer thick paint, glass, and mixed media. I spontaneously apply spirals and swirls of vibrant color that undulate and rotate like waves of an ocean. The spirals and swirls are a recurring motif in my work. These forms are ancient symbols of evolution, growth, and change and reflect the examination of my life. The colors and marks represent the turbulent and constant chaos of life.

In this thesis I make extensive use of the Lyapunov analysis formalism to unravel fundamental mechanisms of instability in two different systems : the Kuramoto model of globally coupled phase-oscillators and the Lorenz 96 (L96) atmospheric "toy" model, portraying the evolution of a physical quantity along a latitude circle. I start by introducing the relevant theoretical background, with special attention on the main tools I have been using throughout this work : Lyapunov Exponents (LEs), which quantify the asymptotic growth rates of infinitesimal perturbations in a system, and by extension, its degree of chaoticity, and Covariant Lyapunov Vectors (CLVs), which indicate the phase space direction (or the geometry) associated with these growth rates. The Kuramoto model is central in the study of synchronization among oscillatory units characterized by their various natural frequencies, but little is known on its chaotic dynamics in the unsynchronized state. I thus investigate the scaling behavior of the first LE, upon different assumptions on the natural frequencies, and make use of educated structural simplifications to analyze the origin of chaos in the finite size model. On the other hand, the L96 model has been devised to gather the main dynamical ingredients of atmospheric dynamics, namely advection, damping, external (solar) forcing and transfers across different scales of motion, in a minimalist and functional way. It features two coupled dynamical layers : the large scale variables, representing synoptic scale atmospheric dynamics, and the small scale variables, faster and more numerous, associated with convective scale dynamics. The core of the study revolves around geometrical properties of CLVs, in the aim of understanding the processes underlying the observed multiscale chaoticity, and an exhaustive study of a non-trivial ensemble of CLVs featuring relevant projection on the slow subspace.