Chaos theory is a branch of modern mathematics concerning the non-linear dynamic systems that are highly sensitive to their initial states. It has extensive real-world applications, such as weather forecasting and stock market prediction. The Lorenz system, defined by three ordinary differential equations (ODEs), is one of the simplest and most popular chaotic models. Historically research has focused on understanding the Lorenz system's mathematical characteristics and dynamical evolution including the inherent chaotic features it possesses. In this thesis, we take a data-driven approach and propose the task of predicting future states of the chaotic system from limited observations. We explore two directions, answering two distinct fundamental questions of the system based on how informed we are about the underlying model. When we know the data is generated by the Lorenz System with unknown parameters, our task becomes parameter estimation (a white-box problem), or the ``inverse'' problem. When we know nothing about the underlying model (a black-box problem), our task becomes sequence prediction. We propose two algorithms for the white-box problem: Markov-Chain-Monte-Carlo (MCMC) and a Multi-Layer-Perceptron (MLP). Specially, we propose to use the Metropolis-Hastings (MH) algorithm with an additional random walk to avoid the sampler being trapped into local energy wells. The MH algorithm achieves moderate success in predicting the $\rho$ value from the data, but fails at the other two parameters. Our simple MLP model is able to attain high accuracy in terms of the $l_2$ distance between the prediction and ground truth for $\rho$ as well, but also fails to converge satisfactorily for the remaining parameters. We use a Recurrent Neural Network (RNN) to tackle the black-box problem. We implement and experiment with several RNN architectures including Elman RNN, LSTM, and GRU and demonstrate the relative strengths and weaknesses of each of these methods. Our results demonstrate the promising role of machine learning and modern statistical data science methods in the study of chaotic dynamic systems. The code for all of our experiments can be found on \url{https://github.com/Yajing-Zhao/}

Submitted by (ana.araujo@ufrpe.br) on 2016-07-06T14:22:48Z No. of bitstreams: 1 Heliovanio Torres Bandeira.pdf: 959027 bytes, checksum: 9873348980adb3c73410a63f86c250d6 (MD5) / Made available in DSpace on 2016-07-06T14:22:48Z (GMT). No. of bitstreams: 1 Heliovanio Torres Bandeira.pdf: 959027 bytes, checksum: 9873348980adb3c73410a63f86c250d6 (MD5) Previous issue date: 2006-06-19 / Ionic channels are formed by one or few protein molecules found in biological membranes and constitute one of the possible ways for the transport of ions through these membranes. These proteins can assume different conformational open and closed states, phenomenon named ion channel kinetics. The transitions from one state to another are dependent on the potential energy barrier that separates them and can be controlled by electric field, ions, chemical substances and other physical agents. The dwell times in which the proteinchannel stays in one these conformational states have been modeled assuming that the process is Markovian. A chaotic model also was proposed for modeling the ion channel kinetics (LIEBOVITCH e TÓTH., 1991).In this work we use the R/S Hurst analysis to test the long-range correlation found in calcium-activated potassium channel kinetics in Leydig cells. The Hurst coefficient H, a parameter that show the memory existent in a kinetic process (NOGUEIRA et al., 1995), was calculated to a calcium-activated potassium channel in Leydig cells recording and it was equal to H = 0,66±0,044 (n=4), disclosing that the system presents a persistent memory. The R/S analysis when applied to the opening and closing dwell time series obtained from ion channel simulated data using a chaotic model was inadequate to describe the long-term correlation previously found in the experimental data. As conclusion, this work shows that: (i) really, opening and closing dwell times for the single calciumactivated potassium channel of Leydig cells present long-term correlation and (ii) the chaotic model, proposed by Liebovitch and Thót (1991), is not adequate to describe the memory found in the kinetic of this channel. / Canais iônicos são compostos de uma ou poucas moléculas de proteínas que se encontram nas membranas biológicas e constituem uma das vias possíveis para o transporte de íons através dessas membranas. Essas proteínas podem assumir diferentes estados conformacionais, abertos e fechados, fenômeno denominado de cinética de canais iônicos. As transições entre os estados cinéticos dos canais dependem das barreiras de energias potenciais que separam esses estados e, que podem ser controladas por campo elétrico, íons, substâncias químicas e outros agentes. Os tempos de permanências dos canais em cada um dos estados conformacionais têm sido modelados assumindo-se que este processo é markoviano. Um modelo caótico também foi proposto para modelar a cinética de canal iônico (LIEBOVITCH e TÓTH, 1991). Neste trabalho utilizamos a análise R/S de Hurst para testar a correlação de longo alcance (memória) na cinética de um canal para potássio ativado por cálcio em células de Leydig. O coeficiente de Hurst H, um parâmetro que mostra a memória existente em um processo cinético (NOGUEIRA et al., 1995), foi calculado para um registro de um canal para potássio ativado por cálcio e foi encontrado um valor de H = 0,66 ± 0,044 (n=4), evidenciando que o sistema apresenta uma memória persistente. A análise R/S aplicada à seqüência temporal de aberturas e fechamentos obtida para um canal iônico simulado por um modelo caótico mostrou que esse modelo é inadequado para descrever a correlação de longo alcance encontrada nos dados experimentais. Como conclusão, este trabalho mostra que: (i) tempos de permanência para aberturas e fechamentos do canal para potássio ativado por cálcio em células de Leydig apresentam correlação de longo alcance (memória);(ii) o modelo caótico, proposto por Liebovitch e Tóth (1991), é inadequado para descrever a memória encontrada na cinética do canal.

This thesis tests the ability of the Lorenz's (2005) chaotic model to simulate predictability curve of the ECMWF model calculated from data over the 1986 to 2011 period and demonstrates similarity of the predictability curves for the Lorenz's model with N = 90 variables. This thesis also tests approximations of predictability curves and their differentials, aiming to correct the ECMWF model estimated parameters and thus allow for estimation of the largest Lyapunov exponent, model error and limit value of the predictability curve. The correction is based on comparing the parameters estimated for the Lorenz's and ECMWF and on comparison with the largest Lyapunov exponent (λ=0,35 day-1 ) and limit value of the predictability curve (E∞=8,2) of the Lorenz's model. Parameters are calculated from approximations made by the Quadratic hypothesis with and without model error, as well as by Logarithmic and General hypotheses and by hyperbolic tangent employing corrections with and without model error. Average value of the largest Lyapunov exponent is estimated to be λ=0,37 day-1 for the ECMWF model, limit values of the predictability curves are estimated with lower theoretically derived values and new approach of calculation of model error based on comparison of models is presented.