Quantifying chaos: practical estimation of the correlation dimension

Be it a physical object or a mathematical model, a nonlinear dynamical system can display complicated aperiodic behavior, or "chaos." In many cases, this chaos is associated with motion on a strange attractor in the system's phase space. And the dimension of the strange attractor indicates the effective number of degrees of freedom in the dynamical system.

In this thesis, we investigate numerical issues involved with estimating the dimension of a strange attractor from a finite time series of measurements on the dynamical system.

Of the various definitions of dimension, we argue that the correlation dimension is the most efficiently calculable and we remark further that it is the most commonly calculated. We are concerned with the practical problems that arise in attempting to compute the correlation dimension. We deal with geometrical effects (due to the inexact self-similarity of the attractor), dynamical effects (due to the nonindependence of points generated by the dynamical system that defines the attractor), and statistical effects (due to the finite number of points that sample the attractor). We propose a modification of the standard algorithm, which eliminates a specific effect due to autocorrelation, and a new implementation of the correlation algorithm, which is computationally efficient.

Finally, we apply the algorithm to chaotic data from the Caltech tokamak and the Texas tokamak (TEXT); we conclude that plasma turbulence is not a low- dimensional phenomenon.

Time and space-resolved measurement of a gas-puff laser-plasma x-ray source

The dynamic interaction processes between a nano-second laser pulse and a gas-puff target, such as those of plasma formation, laser heating, and x-ray emission, have been investigated quantitatively. Time and space-resolved x-ray and optical measurement techniques were used in order to investigate time-resolved laser absorption and subsequent x-ray generation. Efficient absorption of the incident laser energy into the gas-puff target of 17%, 12%, 38%, and 91% for neon, argon, krypton, and xenon, respectively, was shown experimentally. It was found that the laser absorption starts and, simultaneously, soft x-ray emission occurs. The soft x-ray lasts much longer than the laser pulse due to the recombination. Temporal evolution of the soft x-ray emission region was analyzed by comparing the experimental results to the results of the model calculation, in which the laser light propagation through a gas-puff plasma was taken into account. (C) 2003 American Institute of Physics.

Using space geodesy to constrain variations in seismogenic behavior on subduction megathrusts

The concept of seismogenic asperities and aseismic barriers has become a useful paradigm within which to understand the seismogenic behavior of major faults. Since asperities and barriers can be thought of as defining the potential rupture area of large megathrust earthquakes, it is thus important to identify their respective spatial extents, constrain their temporal longevity, and to develop a physical understanding for their behavior. Space geodesy is making critical contributions to the identification of slip asperities and barriers but progress in many geographical regions depends on improving the accuracy and precision of the basic measurements. This thesis begins with technical developments aimed at improving satellite radar interferometric measurements of ground deformation whereby we introduce an empirical correction algorithm for unwanted effects due to interferometric path delays that are due to spatially and temporally variable radar wave propagation speeds in the atmosphere. In chapter 2, I combine geodetic datasets with complementary spatio-temporal resolutions to improve our understanding of the spatial distribution of crustal deformation sources and their associated temporal evolution – here we use observations from Long Valley Caldera (California) as our test bed. In the third chapter I apply the tools developed in the first two chapters to analyze postseismic deformation associated with the 2010 Mw=8.8 Maule (Chile) earthquake. The result delimits patches where afterslip occurs, explores their relationship to coseismic rupture, quantifies frictional properties associated with inferred patches of afterslip, and discusses the relationship of asperities and barriers to long-term topography. The final chapter investigates interseismic deformation of the eastern Makran subduction zone by using satellite radar interferometry only, and demonstrates that with state-of-art techniques it is possible to quantify tectonic signals with small amplitude and long wavelength. Portions of the eastern Makran for which we estimate low fault coupling correspond to areas where bathymetric features on the downgoing plate are presently subducting, whereas the region of the 1945 M=8.1 earthquake appears to be more highly coupled.

Ghosts of order on the frontier of chaos

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderly, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive.

The KAM theory is mute about the disrupted tori, but, for two-dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori," tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.

Control of dynamical systems with temporal logic specifications

This thesis is motivated by safety-critical applications involving autonomous air, ground, and space vehicles carrying out complex tasks in uncertain and adversarial environments. We use temporal logic as a language to formally specify complex tasks and system properties. Temporal logic specifications generalize the classical notions of stability and reachability that are studied in the control and hybrid systems communities. Given a system model and a formal task specification, the goal is to automatically synthesize a control policy for the system that ensures that the system satisfies the specification. This thesis presents novel control policy synthesis algorithms for optimal and robust control of dynamical systems with temporal logic specifications. Furthermore, it introduces algorithms that are efficient and extend to high-dimensional dynamical systems.

The first contribution of this thesis is the generalization of a classical linear temporal logic (LTL) control synthesis approach to optimal and robust control. We show how we can extend automata-based synthesis techniques for discrete abstractions of dynamical systems to create optimal and robust controllers that are guaranteed to satisfy an LTL specification. Such optimal and robust controllers can be computed at little extra computational cost compared to computing a feasible controller.

The second contribution of this thesis addresses the scalability of control synthesis with LTL specifications. A major limitation of the standard automaton-based approach for control with LTL specifications is that the automaton might be doubly-exponential in the size of the LTL specification. We introduce a fragment of LTL for which one can compute feasible control policies in time polynomial in the size of the system and specification. Additionally, we show how to compute optimal control policies for a variety of cost functions, and identify interesting cases when this can be done in polynomial time. These techniques are particularly relevant for online control, as one can guarantee that a feasible solution can be found quickly, and then iteratively improve on the quality as time permits.

The final contribution of this thesis is a set of algorithms for computing feasible trajectories for high-dimensional, nonlinear systems with LTL specifications. These algorithms avoid a potentially computationally-expensive process of computing a discrete abstraction, and instead compute directly on the system's continuous state space. The first method uses an automaton representing the specification to directly encode a series of constrained-reachability subproblems, which can be solved in a modular fashion by using standard techniques. The second method encodes an LTL formula as mixed-integer linear programming constraints on the dynamical system. We demonstrate these approaches with numerical experiments on temporal logic motion planning problems with high-dimensional (10+ states) continuous systems.

General-domain compressible Navier-Stokes solvers exhibiting quasi-unconditional stability and high-order accuracy in space and time

This thesis presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional spatial domains. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of higher-order backward differentiation formulae (BDF) and the alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. In fact this thesis presents, for the first time in the literature, high-order time-convergence curves for Navier-Stokes solvers based on the ADI strategy---previous ADI solvers for the Navier-Stokes equations have not demonstrated orders of temporal accuracy higher than one. An extended discussion is presented in this thesis which places on a solid theoretical basis the observed quasi-unconditional stability of the methods of orders two through six. The performance of the proposed solvers is favorable. For example, a two-dimensional rough-surface configuration including boundary layer effects at Reynolds number equal to one million and Mach number 0.85 (with a well-resolved boundary layer, run up to a sufficiently long time that single vortices travel the entire spatial extent of the domain, and with spatial mesh sizes near the wall of the order of one hundred-thousandth the length of the domain) was successfully tackled in a relatively short (approximately thirty-hour) single-core run; for such discretizations an explicit solver would require truly prohibitive computing times. As demonstrated via a variety of numerical experiments in two- and three-dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, useful stability properties, limited dispersion, and high parallel efficiency.

Ultrafast temporal pattern generation and recognition

Ultrafast temporal pattern generation and recognition with femtosecond laser technology is presented, analyzed, and experimentally implemented. Ultrafast temporal pattern generation and recognition are realized by taking advantage of two well-known techniques: the space-time conversion technique and the ultrafast pulse measurement technique. Here the temporal pattern for the designed multiple pulses, optimized with a preassumed Gaussian spectral distribution of an ultrashort pulse, is described. With the simulation of a Gaussian spectral distribution, we realize that the uniformity of the generated multiple ultrafast temporal pulses is relevant to the repeated number of modulation periods in the mask in the spectral plane. Moreover, the change of Gaussian spectral phases with the wavelengths in the modulated phase plate is considered. Experiments of ultrafast temporal pattern recognition by the frequency-resolved optical gating (FROG) characterization technique are also given. (C) 2004 Society of Photo-Optical Instrumentation Engineers.

Avaliação espaço-temporal do grau de trofia em lagoas costeiras da cidade do Rio de Janeiro: aplicação de índices de estado trófico

Os diversos tipos de lançamentos de cargas poluidoras podem tornar a qualidade da água inadequada para os usos benéficos previstos. Entre as causas desta queda na qualidade da água, pode-se citar a eutrofização. O grau de trofia pode ser estimado através da utilização de índices, destacando-se o Índice do Estado Trófico desenvolvido por Carlson (1977) e o Índice do Estado Trófico desenvolvido por Carlson, modificado por Toledo et al. (1984). O presente estudo teve como principal objetivo analisar a variação espaço-temporal entre o período de 1980 a 2008, a fim de verificar o estado trófico das águas da Lagoa Rodrigo de Freitas (RJ) e do Complexo Lagunar de Jacarepaguá (RJ), através do uso destes índices. As variáveis utilizadas foram clorofila-a, transparência da água e fósforo total, e para o IETm, foi acrescido a variável ortofosfato dissolvido. Os resultados obtidos indicaram que tanto a Lagoa Rodrigo de Freitas, quanto o Complexo Lagunar de Jacarepaguá estão classificados como ambientes hipereutróficos no índice de Carlson e como eutróficos no índice modificado por Toledo. Entretanto, estes corpos dágua encontram-se em fases distintas. Para a Lagoa Rodrigo de Freitas, foi observada uma melhora na qualidade de suas águas nas últimas décadas, já para o Complexo Lagunar de Jacarepaguá, foi constatado uma piora significativa da qualidade de suas águas, principalmente nesta última década. A aplicação dos índices do estado trófico demonstrou-se uma ferramenta de avaliação do grau de trofia dos corpos dágua bastante prática, de fácil interpretação e divulgação dos dados obtidos a partir de um monitoramento sistemático

Caracterização espaço-temporal de plumas de sedimentos por sensoriamento remoto: um estudo de caso na foz do rio Paraíba do Sul.

Sedimentos em suspensão representam um dos principais fatores que afetam a qualidade dos sistemas aquáticos no mundo; influenciam os processos geomórficos de construção da paisagem e podem indicar problemas de erosão e perda de solo na bacia hidrográfica contribuinte. O seu monitoramento espacial e temporal é fundamental nas atividades de gestão ambiental de áreas costeiras. Nesse sentido, a hipótese básica desta pesquisa é que o padrão espacial e temporal de plumas de sedimentos costeiras associado ao regime hidrológico do rio pode ser caracterizado a partir de imagens orbitais de média resolução espacial. Para comprová-la, elegeu-se a foz do rio Paraíba do Sul como área de estudo para definição e teste metodológico, e formulou-se como principal objetivo mapear qualitativamente a pluma costeira deste rio a partir de imagens Landsat 5 e CBERS-2, ao longo do período compreendido entre 1985 e 2007. As datas avaliadas foram criteriosamente definidas através de três estratégias de análise, totalizando cinqüenta imagens. Pesquisa bibliográfica e avaliação da resposta espectral da feição de interesse nas imagens selecionadas consistiram nas etapas principais da definição da metodologia. As plumas foram então identificadas, mapeadas e extraídas; posteriormente, suas características espaciais e temporais foram analisadas por intermédio de sistemas de informação geográfica e avaliadas em conjunto com dados históricos de vazão. Os resultados indicam que a banda do vermelho forneceu uma melhor discriminação interna da pluma, sendo, portanto, utilizada como base para as análises realizadas neste trabalho. Com exceção do procedimento de correção atmosférica, a metodologia proposta consiste na utilização de técnicas simples de processamento digital de imagens, baseadas na integração de técnicas semi-automáticas e de análise visual. A avaliação do padrão dos sedimentos e dos mapas temáticos qualitativos de concentração de sedimentos em suspensão indica a forte diferenciação existente entre cenários representativos de épocas de cheia e seca do rio. Análises espaciais do comportamento da pluma contribuem ainda para um maior conhecimento do espaço geográfico, fornecendo subsídios aos mais variados setores do planejamento e gestão ambiental.

Aperfeiçoamento da previsão global em séries temporais caóticas

A previsão de valores futuros em séries temporais produzidas por sistemas caóticos pode ser aplicada em diversas áreas do conhecimento como Astronomia, Economia, Física, Medicina, Meteorologia e Oceanografia. O método empregado consiste na reconstrução do espaço de fase seguido de um termo de melhoria da previsão. As rotinas utilizadas para a previsão e análise nesta linha de pesquisa fazem parte do pacote TimeS, que apresenta resultados encorajadores nas suas aplicações. O aperfeiçoamento das rotinas computacionais do pacote com vistas à melhoria da acurácia obtida e à redução do tempo computacional é construído a partir da investigação criteriosa da minimização empregada na obtenção do mapa global. As bases matemáticas são estabelecidas e novas rotinas computacionais são criadas. São ampliadas as possibilidades de funções de ajuste que podem incluir termos transcendentais nos componentes dos vetores reconstruídos e também possuir termos lineares ou não lineares nos parâmetros de ajuste. O ganho de eficiência atingido permite a realização de previsões e análises que respondem a perguntas importantes relacionadas ao método de previsão e ampliam a possibilidade de aplicações a séries reais.

Variabilidade espacial e temporal das concentrações de clorofila na Baía de Guanabara (RJ) utilizando imagens MERIS e dados in situ

Este trabalho teve como objetivo principal implementar um algoritmo empírico para o monitoramento do processo de eutrofização da Baía de Guanabara (BG), Rio de Janeiro (RJ), utilizando dados de clorofila-a coletados in situ e imagens de satélite coletadas pelo sensor MERIS, a bordo do satélite ENVISAT, da Agência Espacial Européia (ESA). Para a elaboração do algoritmo foi utilizada uma série histórica de clorofila-a (Out/2002 a Jan/2012) fornecida pelo Laboratório de Biologia Marinha da UFRJ, que, acoplada aos dados radiométricos coletados pelo sensor MERIS em datas concomitantes com as coletas in situ de clorofila-a, permitiu a determinação das curvas de regressão que deram origem aos algorítmos. Diversas combinações de bandas foram utilizadas, com ênfase nos comprimentos de onda do verde, vermelho e infra-vermelho próximo. O algoritmo escolhido (R = 0,66 e MRE = 77,5%) fez uso dos comprimentos de onda entre o verde e o vermelho (665, 680, 560 e 620 nm) e apresentou resultado satisfatório, apesar das limitações devido à complexidade da área de estudo e problemas no algoritmo de correção atmosférica . Algorítmos típicos de água do Caso I (OC3 e OC4) também foram testados, assim como os algoritmos FLH e MCI, aconselhados para águas com concentrações elevadas de Chl-a, todos com resultados insatisfatório. Como observado por estudos pretéritos, a Baia de Guanabara possui alta variabilidade espacial e temporal de concentrações de clorofila-a, com as maiores concentrações no período úmido (meses: 01, 02, 03, 10, 11 12) e nas porções marginais (~ 100 mg.m-3), particularmente na borda Oeste da baia, e menores concentrações no período seco e no canal principal de circulação (~ 20 mg.m-3). O presente trabalho é pioneiro na construção e aplicação de algoritmos bio-óptico para a região da BG utilizando imagens MERIS. Apesar dos bons resultados, o presente algorítmo não deve ser considerado definitivo, e recomenda-se para trabalhos futuros testar os diferentes modelos de correção atmosférico para as imagens MERIS.

Breathers and thermal relaxation as a temporal process: A possible way to detect breathers in experimental situations

Breather stability and longevity in thermally relaxing nonlinear arrays is investigated under the scrutiny of the analysis and tools employed for time series and state reconstruction of a dynamical system. We briefly review the methods used in the analysis and characterize a breather in terms of the results obtained with such methods. Our present work focuses on spontaneously appearing breathers in thermal Fermi-Pasta-Ulam arrays but we believe that the conclusions are general enough to describe many other related situations; the particular case described in detail is presented as another example of systems where three incommensurable frequencies dominate their chaotic dynamics (reminiscent of the Ruelle-Takens scenario for the appearance of chaotic behavior in nonlinear systems). This characterization may also be of great help for the discovery of breathers in experimental situations where the temporal evolution of a local variable (like the site energy) is the only available/measured data. © 2005 American Institute of Physics.

The temporal and spatial evaluation on China's agricultural policy output since 1978

Since reform and opening up, how much contribution has China's implementation of new agricultural policy made to agricultural output? This paper is trying to establish an agricultural policy output econometric model for doing a quantitative analysis of China's new agricultural policy. The results show that China's agricultural policies on agricultural output have an average contribution rate of about 7% since 1978, which is consistent with the OECD's basic forecast. There are obvious temporal and spatial differences. Generally speaking, we can divide the contribution of agricultural policy into three periods, which are the start-up phase from 1978 to 1991 (14 years), the stationary phase from 1992 to 2002 (11 years) and the rising phase from 2003 to 2008 (6 years). In space, the contribution of agricultural policy underwent a process from the all-low in the start-up phase, the gradual increase in the stationary phase to the all-high in the rising phase. Northern and western regions are more sensitive to policies. There are three major factors that can affect the contribution of regional agricultural policies, which are the process of national industrialization strategy, terrain and the level of local finance.

Articulated Pose Estimation in a Learned Smooth Space of Feasible Solutions

A learning based framework is proposed for estimating human body pose from a single image. Given a differentiable function that maps from pose space to image feature space, the goal is to invert the process: estimate the pose given only image features. The inversion is an ill-posed problem as the inverse mapping is a one to many process. Hence multiple solutions exist, and it is desirable to restrict the solution space to a smaller subset of feasible solutions. For example, not all human body poses are feasible due to anthropometric constraints. Since the space of feasible solutions may not admit a closed form description, the proposed framework seeks to exploit machine learning techniques to learn an approximation that is smoothly parameterized over such a space. One such technique is Gaussian Process Latent Variable Modelling. Scaled conjugate gradient is then used find the best matching pose in the space of feasible solutions when given an input image. The formulation allows easy incorporation of various constraints, e.g. temporal consistency and anthropometric constraints. The performance of the proposed approach is evaluated in the task of upper-body pose estimation from silhouettes and compared with the Specialized Mapping Architecture. The estimation accuracy of the Specialized Mapping Architecture is at least one standard deviation worse than the proposed approach in the experiments with synthetic data. In experiments with real video of humans performing gestures, the proposed approach produces qualitatively better estimation results.

Real-Time Spatio-Temporal Query Processing in Mobile Ad-Hoc Sensor Networks

Personal communication devices are increasingly equipped with sensors that are able to collect and locally store information from their environs. The mobility of users carrying such devices, and hence the mobility of sensor readings in space and time, opens new horizons for interesting applications. In particular, we envision a system in which the collective sensing, storage and communication resources, and mobility of these devices could be leveraged to query the state of (possibly remote) neighborhoods. Such queries would have spatio-temporal constraints which must be met for the query answers to be useful. Using a simplified mobility model, we analytically quantify the benefits from cooperation (in terms of the system's ability to satisfy spatio-temporal constraints), which we show to go beyond simple space-time tradeoffs. In managing the limited storage resources of such cooperative systems, the goal should be to minimize the number of unsatisfiable spatio-temporal constraints. We show that Data Centric Storage (DCS), or "directed placement", is a viable approach for achieving this goal, but only when the underlying network is well connected. Alternatively, we propose, "amorphous placement", in which sensory samples are cached locally, and shuffling of cached samples is used to diffuse the sensory data throughout the whole network. We evaluate conditions under which directed versus amorphous placement strategies would be more efficient. These results lead us to propose a hybrid placement strategy, in which the spatio-temporal constraints associated with a sensory data type determine the most appropriate placement strategy for that data type. We perform an extensive simulation study to evaluate the performance of directed, amorphous, and hybrid placement protocols when applied to queries that are subject to timing constraints. Our results show that, directed placement is better for queries with moderately tight deadlines, whereas amorphous placement is better for queries with looser deadlines, and that under most operational conditions, the hybrid technique gives the best compromise.