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@PhDThesis{Ferreira:2014:DeSiFa,
               author = "Ferreira, Maria Teodora",
                title = "Detec{\c{c}}{\~a}o da sincroniza{\c{c}}{\~a}o de fase em 
                         sistemas ca{\'o}ticos por meio da transformada wavelet complexa 
                         dual–tree",
               school = "Instituto Nacional de Pesquisas Espaciais (INPE)",
                 year = "2014",
              address = "S{\~a}o Jos{\'e} dos Campos",
                month = "2014-06-24",
             keywords = "sincroniza{\c{c}}{\~a}o de fase, transformada wavelet complexa 
                         dual-tree, sistemas ca{\'o}ticos, an{\'a}lise de s{\'e}ries 
                         temporais, phase synchronization, dual-tree complex wavelet 
                         transform, chaotic systems, time series analysis.",
             abstract = "Neste trabalho {\'e} proposto um novo m{\'e}todo para calcular a 
                         fase de sistemas ca{\'o}ticos e de conjuntos de dados 
                         experimentais, a fim de analisar o fen{\^o}meno de 
                         sincroniza{\c{c}}{\~a}o de fase. A sincroniza{\c{c}}{\~a}o de 
                         fase ocorre, principalmente, quando se tem um acoplamento fraco 
                         entre os sistemas ocasionando um travamento na fase enquanto as 
                         amplitudes permanecem n{\~a}o correlacionadas. O m{\'e}todo 
                         proposto, denominado \emph{wavelet dt-cwt}, {\'e} baseado na 
                         utiliza{\c{c}}{\~a}o da transformada \emph{wavelet} discreta 
                         complexa \emph{dual-tree}, a qual {\'e} uma transformada 
                         \emph{wavelet} complexa quase-ortogonal, com algoritmos 
                         r{\'a}pidos, possui boa invari{\^a}ncia a deslocamentos e 
                         redund{\^a}ncia limitada. A aplicabilidade do m{\'e}todo 
                         \emph{wavelet dt-cwt} {\'e} testada em: cinco diferentes 
                         experimentos envolvendo o sistema de R{\~o}ssler; dois 
                         experimentos envolvendo o sistema de Lorenz; no Modelo de Kuramoto 
                         imerso no Plano e na Faixa de M{\'o}ebius e, em quatro diferentes 
                         conjuntos de dados experimentais. Os resultados obtidos com o 
                         m{\'e}todo mostram-se eficientes em rela{\c{c}}{\~a}o aos 
                         m{\'e}todos j{\'a} existentes para este fim, devido a seu baixo 
                         custo computacional, da ordem de \emph{2N} com N sendo o 
                         n{\'u}mero de pontos da s{\'e}rie temporal, quando comparado com 
                         a transformada \emph{wavelet} cont{\'{\i}}nua, a qual apresenta 
                         um custo da ordem de \$N^{2}\$. A efici{\^e}ncia do m{\'e}todo 
                         proposto tamb{\'e}m se destaca no fato do mesmo n{\~a}o exigir a 
                         reconstru{\c{c}}{\~a}o do atrator no espa{\c{c}}o de estado, 
                         sendo que, {\'e} suficiente apenas uma vari{\'a}vel de cada 
                         sistema em estudo ou a s{\'e}rie temporal experimental; devido a 
                         possibilidade de ser aplicado em s{\'e}ries temporais com grande 
                         n{\'u}mero de pontos; o fato de ser robusto a moderados 
                         n{\'{\i}}veis de ru{\'{\i}}do aditivo e sua 
                         localiza{\c{c}}{\~a}o acurada nas mudan{\c{c}}as de fase. 
                         ABSTRACT: In this work, it is proposed a method to calculate the 
                         phase of chaotic systems and of sets of experimental data in arder 
                         to analyze the phenomenon of phase synchronization. The phase 
                         synchronization mainly occurs when there is a coupling weak 
                         between the systems causing a locking in the phases while the 
                         amplitudes remain uncorrelated. The proposed method,called wavelet 
                         dt-cwt, is based on the use of dual-tree complex wavelet 
                         transform, which is a quasi-orthogonal complex wavelet transform, 
                         with fast algarithms, has good invariance to shifts and limited 
                         redundancy. The applicability of the \emph{wavelet dt-cwt} method 
                         is tested in: five different experiments involving the system of 
                         R{\^o}ssler: two experiments involving the Lorenz system; the 
                         Kuramoto model immersed in the Plan and the Moebius Strip and with 
                         and without noise, and four different sets of experimental data. 
                         The results show the method is efficient compared with existing 
                         methods for this purpose, due to its low computational cost - on 
                         the order of 2N with N being the number of points in the time 
                         series - when compared with the continuous wavelet transform, 
                         which presents a cost the order of \$N^{2}\$. The efficiency of 
                         the proposed method also stands out in the fact that the same does 
                         not require the reconstruction of the attractor in the state 
                         space, being that, is sufficient a single variable of each system 
                         under study or experimental time series; due to the possibility of 
                         being applied in series with a large number of points; the fact 
                         that it is robust to moderate leveis of additive noise and its 
                         accurate location on the phase changes.",
            committee = "Quiles, Marcos Gon{\c{c}}alves (presidente) and Macau, Elbert 
                         Einstein Nehrer (orientador) and Domingues, Margarete Oliveira 
                         (orientador) and Castro, Joaquim Jos{\'e} Barroso de and 
                         Carvalho, Ricardo Egydio de and Viana, Ricardo Luiz",
         englishtitle = "Detecting phase synchronization in chaotic systems by dual-tree 
                         complex wavelet transform",
             language = "pt",
                pages = "202",
                  ibi = "8JMKD3MGP5W34M/3GESL58",
                  url = "http://urlib.net/rep/8JMKD3MGP5W34M/3GESL58",
           targetfile = "publicacao.pdf",
        urlaccessdate = "22 set. 2020"
}


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