<p>In 1998 the Swedish mathematician Warwick Tucker used rigorous interval arithmetic and normal form theory to prove the existence of a strange attractor in the Lorenz system. In large parts, that proof consists of computations implemented and performed on a computer. This thesis is an independent numerical verification of the result obtained by Warwick Tucker, as well as a study of a higher-dimensional system of ordinary differential equations introduced by the Swedish physicist Lennart Stenflo.</p><p>The same type of mapping data as Warwick Tucker obtained is calculated here via a combination of numerical integration, solving optimisation problems and a coordinate change that brings the system to a normal form around the stationary point in the origin. This data is collected in a graph and the problem of determining the existence of a strange attractor is translated to a few graph theoretical problems. The end result, after the numerical study, is a support for the conclusion that the attractor set of the Lorenz system is a strange attractor and also for the conclusion that the Lorenz-Stenflo system possesses a strange attractor.</p>

In 1998 the Swedish mathematician Warwick Tucker used rigorous interval arithmetic and normal form theory to prove the existence of a strange attractor in the Lorenz system. In large parts, that proof consists of computations implemented and performed on a computer. This thesis is an independent numerical verification of the result obtained by Warwick Tucker, as well as a study of a higher-dimensional system of ordinary differential equations introduced by the Swedish physicist Lennart Stenflo. The same type of mapping data as Warwick Tucker obtained is calculated here via a combination of numerical integration, solving optimisation problems and a coordinate change that brings the system to a normal form around the stationary point in the origin. This data is collected in a graph and the problem of determining the existence of a strange attractor is translated to a few graph theoretical problems. The end result, after the numerical study, is a support for the conclusion that the attractor set of the Lorenz system is a strange attractor and also for the conclusion that the Lorenz-Stenflo system possesses a strange attractor. / QC 20101007

This thesis examines how novel invariants called the "stability index" as proposed by Podvigina and Ashwin can be used to characterize the local geometry of riddled basins of attraction for both skew and non-skew product systems. In particular, it would be interesting to understand how the stability index behaves on the basin boundary between multiple basins of attraction. Then we can ask this question: How can we identify when a basin is riddled? To answer this, we present three models with the presence of riddled basins. In the first model, we present a skew product system of a simple example of a piecewise linear map. We prove that the riddled basin occurs within a certain range of parameter and calculate the stability index analytically for this map. Our results for the stability index at a point show that for Lebesgue almost all points in the map, the index is positive and for some points the index may be negative. We verify these results with our numerical computation for this index. We also make a corollary claiming that the formula for the stability index at a point can be expressed in terms of the stability index for an attractor and Lyapunov exponents for this map. This suggests that this index could be useful as a diagnostic tool to study bifurcation of the riddled basins of attraction. In the second model, we refer to a skew product map studied by Keller. Previously, Keller computed the stability index for an attractor in his map whereas in this thesis, we use an alternative way to compute the index; that is on the basins of attraction for Keller's map, found by inverting his map. Using the same map, we also verify maximum and minimum measures as obtained in his paper by studying Birkhoff averages on periodic points of Markov map in his system. We also conjecture result by Keller and Otani on the dimension of zero sets of invariant graph (i.e. basin boundary) that appears in Keller's map to a complete range of a parameter in the map. The last model is a non-skew product map which is also has a riddled basin. For this map, we compute the stability index for an attractor on the baseline of the map. The result indicates that the index is positive for Lebesgue almost all points whenever the riddled basin occurs.

O propósito deste trabalho é estudar a existência de solução periódica para problemas de oscilação não linear de barras submetidas a forças periódicas. Estudaremos concretamente dois problemas, que serão interpretados como equações diferenciais abstratas de segunda ordem cuja classe foi considerada em Ceron e Lopes [1]. Para garantir a existência de solução periódica dos problemas considerados, mostraremos que a aplicação de Poincaré S é limitada dissipativa e alfa-contração. Isso garante a existência de um atrator invariante compacto e a existência de um ponto fixo de S, o que é equivalente a existência da solução periódica. / Our aim in this work is to study the existence of periodic solution to oscillation in nonlinear problems of beams submitted to periodic forcing. We will study concretely two problems, which can be interpreted as an abstract second order diferential equation studied by Ceron and Lopes [1]. Our intention is to prove the existence of periodic solution to these problems. To this end, we will show that the Poincaré map S is uniform ultimately bounded and alpha-contraction. Thus we have the existence of invariant compact attractor, therefore S have a fixed point, which is equivalent the existence of a periodic solution.

Thesis advisor: Julie H. Mortimer / This dissertation consists of two independent chapters on pricing and consumer demand in the retail sector. In chapter 1 develop an empirical model of Consumer Supermarket Choice that enables identification of heterogeneous consumer travel costs and is suitable for a wide range of policy experiments and the study of local competition. Chapter 2 is a theoretical investigation on pricing patterns in multi-product retail markets, when boundedly rational consumers' choice of a store is based on the price and valuation of a subset of goods. Estimation of demand systems in spatially differentiated retail markets is fundamental for understanding local competition and the impact of policy changes. It is also challenging, because shopping decisions consist of multiple dimensions: when to shop, where to shop and what to buy. In chapter 1 I develop an empirically tractable model of store choice in the supermarket industry and provide a way to identify consumers' heterogeneous travel costs without imposing restrictions on bundle choice. Using micro level data on a small market in New England, I estimate demand for stores using both a moment inequality approach and standard discrete choice techniques. I specify utility as a function of both store and bundle characteristics, and control for the endogeneity of expenditure on the bundle. I use the estimates of the discrete choice model to evaluate the welfare impact of 1) the closing of each individual store in the market and 2) the relocation of one of the stores. I find that travel costs are heterogeneous and marginally decreasing; that people like to shop at stores that are close, but also like to shop at multiple stores. Furthermore, people value stores differently (across consumers and shopping occasion) and trade off additional travel time for better store characteristics; utility differentials in preference for stores correspond to a distance ranging between zero and up to 3.3 miles. Variation in demand and substitution patterns across stores are explained by differences in store characteristics and by the shopping habits and geographic distribution of heterogenous consumers. Changes in market structure, like store entry and exit can have significant impact on consumer welfare. For example, removal on one of the stores results in a loss in CS that ranges between 8% and 44%. The assumption of rationality in retail shopping decisions appears very problematic when stores sell thousands of products and frequently vary their assortments and prices. Consumers are typically uncertain about prices at different stores and for a consumer to consider the entire distribution of bundles and prices might be a far too complex decision process. Furthermore, models with rational consumers are incapable of fully explaining important features of retail markets such as price dispersion, advertising and leader pric- ing. In chapter 2 I attempt to characterize optimal pricing by multi-product retailers when imperfectly informed consumers buy more than one product. The distinctive feature of the model is that there are two relevant moments to all purchase decisions. First, the choice of a store to visit, and second, the choice of the items to purchase. While consumers might rationally choose a store to best meet their specific needs and desires, the choice of the items to purchase is made only once in a store. Whether guided by impulse, contingent and unforeseen needs or in-store learning about a product, consumers often end up buying additional products which can generate higher profits for the stores. To examine the implications on retail pricing of this kind of behavior, I depart from a standard rational setup and introduce the concept of attractor goods. Using an an approach similar to that found in Osborne and Rubinstein (1998) and Spiegler (2006) I consider boundedly rational con- sumers whose choice between stores is based solely and entirely on the price and valuation of a subset of goods, the attractors. I show that retailer's pricing decisions have to take into account not only the direct effect of prices on a product's demand but also the effect on the demand for the other products sold in the store. The optimal pricing schedule will be a decreasing function of the goods' attractiveness, and pricing below marginal cost might be optimal for some goods. The model provides a rationale for the strategy of loss leader pricing and offers an intuitive explanation to countercyclical markups. / Thesis (PhD) — Boston College, 2013. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.

Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, em invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system. Considering weak tracking for a nonautonomous Rossler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor.

A partir de uma aplicação da Forma Canônica de Jordan, construímos uma base para o espaço atrator para operadores quânticos de dimensão finita. Essa base é formada pelos autoespaços correspondentes a autovalores de módulo 1. Com essa construção, descrevemos o comportamento da dinâmica assint otica dos operadores quânticos, obtendo assim, o resultado principal do texto. A dinâmica depende dos vetores duais, cuja definição não é feita a partir de uma forma explicita, mas por propriedades relacionadas ao traço. Investigando propriedades dos operadores estritamente positivos, definimos um produto interno que relaciona o produto interno de Hilbert-Schmidt com um operador estritamente positivo. Com isso, obtemos uma forma explícita para os vetores duais. / From an application of the Jordan Canonical Form, we construct a basis for the attractor space for quantum operations of nite dimension. This basis is formed by eigenspaces corresponding to eigenvalues of modulus 1. With this construction, we describe the behavior of the asymptotic dynamics of the quantum operations, thus obtaining the main result of the text. The dynamics depends on the dual vectors whose de nition is not made in an explicit form, but by properties related to the trace. Investigating the properties of strictly positive operators, we de ne an inner product that relates the Hilbert-Schmidt inner product with a strictly positive operator. Thus, we have an explicit form for the dual vectors.

Anti-control of chaos has attracted a lot of attention recently due to its potential applications in science and engineering. How to generate useful chaos that is also practically implementable and useful is a current focus of research. This research aims at developing new chaos generation schemes which demonstrate complex dynamical behaviours using simple linear systems with hysteresis function series. A continuous-time linear unstable second-order system with a feedback of hysteresis function is first proposed for generating chaos. The design for chaos generation is studied theoretically. A Poincaré map is used to demonstrate the dynamical behaviour of the system. The existence and the analytic solution of the limit cycle that bounds the basin of attraction of the chaotic attractor are derived. Conditions for the existence of chaotic attractors are studied. A hysteresis based system with a maximum chaotic stability margin is designed. Second, systematic methods for generating 1D n-scroll chaotic attractors in the directions of the state variables and 2D nxm-grid scroll chaotic attractors in the phase plane via continuous-time linear unstable second-order systems with a feedback of hysteresis function series are proposed. Furthermore, systematic methods for generating 1D n-scroll, 2D nxm-grid scroll and 3D nxmxl-space scroll chaotic attractors via continuous-time linear unstable third-order systems using hysteresis function series feedback are also presented in this thesis. Simulation results are presented to demonstrate effectiveness of the schemes. It is shown that the multi-scroll chaos generation systems can be represented in Lur'e form, and as a result it may be used within synchronization schemes for secure communication. Third, the limit cycle that bounds the basin of attraction in the multi-scroll chaos generation with second-order systems case is studied. The relationship of the size of the basin of attraction with the numbers of hysteresis function series is studied. The multi-scroll chaos generation mechanism is then further explored by analyzing the system trajectories; the switching boundaries, switching rules and the trajectories on each subspace. The chaotic behaviours are confirmed theoretically and it is proved that a non-ordinary attractor exists in the multi-scroll chaotic attractor of the second-order systems case. The abundant dynamical behaviour of the multi-scroll chaos generation systems using different hysteresis feedback are demonstrated. A double-hysteresis function, which is the superimposition of two basic hysteresis functions, is proposed for the implementation of the hysteresis based chaotic system. In this design, the double-hysteresis block and its series are constructed via a systematic method. The ideal hysteresis function series can be implemented easily with the proposed double-hysteresis function. The number of scroll attractors can be designed arbitrarily, and the multi-scroll chaotic attractors can be located anywhere and cover any chosen area of the phase plane. The circuitry implementation for generating 1D n-scroll, 2D nxm-grid scroll chaotic attractors with linear second-order systems and hysteresis function series is given. And the oscilloscope illustrated waveforms which included as many as 9x9=81 scrolls chaotic attractor are presented. The experimental results confirmed the theoretical analysis very well and validated the effectiveness as well as the feasibility of the proposed multi-scroll chaos generation schemes. This research may find potential engineering applications in areas such as digital coding and image processing, etc.

Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

A computational technique is introduced to reveal the complex intrinsic structure of homoclinic and heteroclinic bifurcations in a chaotic dynamical system. This technique is applied to several Lorenz-like systems with a saddle at the center, including the Lorenz system, the Shimizu-Morioka model, the homoclinic garden model, and the laser model. A multi-fractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities is explored on a bi-parametric plane of those dynamical systems. Also a great detail is explored in the Shimizu-Morioka model as an example. The technique is also applied to a re exion symmetric dynamical system with a saddle-focus at the center (Chua's circuits). The layout of the homoclinic bifurcations near the primary one in such a system is studied theoretically, and a scalability ratio is proved. Another part of the dissertation explores the intrinsic mechanisms of escape in a reciprocally inhibitory FitzHugh-Nagumo type threecell network, using the phase-lag technique. The escape network can produce phase-locked states such as pace-makers, traveling-waves, and peristaltic patterns with recurrently phaselag varying.