Órbitas periódicas instáveis, sincronização de caos e transientes em redes hiperbólicas

Orientador: Prof. Dr. Ricardo Luiz VianaCoorientador: Prof. Dr. Sandro Eloy de Souza PintoDissertação (mestrado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Física. Defesa: Curitiba, 26/02/2009Inclui referências : f. 94-95Resumo: As órbitas periódicas instáveis são o esqueleto sobre o qual a dinâmica caótica é construída. Imerso em uma sela ou atrator caótico, existe um conjunto infinito enumerável de tais órbitas. Esse conjunto infinito contável, embora possua medida de Lebesgue nula, suporta a medida natural dos atratores caóticos e o decaimento da medida para selas caóticas. Relações exatas e expansões em séries convergentes para quantidades dinâmicas fundamentais, tais como a entropia topológica e a taxa de escape, podem ser construídas em termos das órbitas periódicas instáveis para sistemas hiperbólicos. Nesse trabalho, um sistema caótico de alta dimensão foi analisado pelas ferramentas da teoria das órbitas periódicas. Mais especificamente, uma rede composta de diversos mapas de Bernoulli acoplados foi estudada. A hiperbolicidade dessa rede pode ser provada e, portanto, o formalismo das órbitas periódicas formalmente pode ser aplicado. A intensidade do acoplamento de cada par de sítios depende da distância entre eles na rede de acordo com uma lei de potência. As trajetórias típicas dessa rede podem exibir sincronização, desde que o estado sincronizado, dado por uma variedade unidimensional S, seja estável. De modo a determinar a estabilidade local dessa variedade, o espectro de Lyapunov das trajetórias sincronizadas foi avaliado. A análise da estabilidade global desse estado foi calculada pela expressão para a medida natural em termos do conjunto denso de órbitas periódicas instáveis imersas em S. Para os casos em que S é globalmente estável, experimentos numéricos para o tempo de sincronização foram realizados. O comportamento errático e instável observado nas trajetórias do estado dessincronizado, bem como os tempos de sincronização médios muito longos, estabeleceram a hipótese sobre a existência de uma sela caótica G no espaço de fase da rede. O decaimento exponencial temporal típico no número de trajetórias capturadas pela sela foi verificado, para o qual a taxa de escape pode ser calculada. As órbitas periódicas instáveis de G foram numericamente determinadas pela implementação do método de transformação de estabilização e, então, a topologia da sela caótica pôde ser analisada. Uma vez que uma expressão analítica biunívoca para a associação simbólica às órbitas periódicas foi encontrada, a expansão em ciclos primos pôde ser formalmente aplicada e a taxa de escape de G calculada. A exatidão dos resultados obtidos, quando comparados com o ajuste do decaimento exponencial das trajetórias, formalmente comprova a existência da sela caótica G . Consequentemente, o tempo de sincronização médio é associado ao tempo de vida média da sela caótica imersa no espaço de fase da rede. Palavras-chave: órbitas periódicas instáveis, sincronização de caos, sistemas hiperbólicos, tempo de sincronização, transientes caóticos, expansão em ciclos primos.Abstract: Unstable periodic orbits are the skeleton upon which the chaotic dynamics is built. Embedded in a chaotic saddle or attractor, there is a denumerable infinite set of such orbits. This countable infinite set, although has zero Lebesgue measure, supports the natural measure of chaotic attractors and the measure decay for chaotic saddles. Exact relationships and convergent series expansions for fundamental dynamical quantities, such as topological entropy and escape rate, can be constructed in terms of unstable periodic orbits for hyperbolic systems. In this work, a high-dimensional chaotic system was analyzed by the tools of periodic orbit theory. More specifically, a lattice composed by several coupled Bernoulli maps was studied. This lattice hyperbolicity can be proved and, therefore, the periodic orbit formalism formally can be applied. The coupling intensity of each pair of sites depends on the lattice distance between them in a power-law fashion. Typical trajectories of such lattice can exhibit synchronization, provided that the synchronization state, given by a one-dimensional manifold S, is stable. In order to determine the local stability of this manifold, the Lyapunov expectra of synchronized trajectories was evaluated. The global stability analysis of such state was computed by the natural measure expression in terms of the dense set of unstable periodic orbits embedded inS. For the situations in whichS is globally stable, numerical experiments for the synchronization time were performed. The observed unstable and erratic behavior of the desynchronized state trajectories as well as the very long average synchronization times establish the hypothesis about the existence of a chaotic saddle G in the lattice phase space. The typical temporal exponential decay in the number of trajectories trapped by the saddle was verified, for which the escape rate could be computed. The unstable periodic orbits of G were numerically determined by the implementation of the stabilization transformation procedure and, therefore, the topology of the chaotic saddle could be analyzed. Since an analytical expression for the one-to-one symbolic association to the periodic orbits was found, the prime cycle expansion could be formally applied and the escape rate of G computed. The correctness of the found results, when compared with the trajectories exponential decay fit, formally prove the existence of the chaotic saddle G . Consequently, the average synchronization time is associated with the average lifetime of a chaotic saddle embedded in the lattice phase space. Keywords: unstable periodic orbits, chaos synchronization, hyperbolic systems, synchronization time, chaotic transients, prime cycle expansion

Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system

Chaotic dynamical systems with two or more attractors lying on invariant subspaces may, provided certain mathematical conditions are fulfilled, exhibit intermingled basins of attraction: Each basin is riddled with holes belonging to basins of the other attractors. In order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest (two-species competition with extinction) we have characterized quantitatively the intermingled basins using periodic-orbit theory and scaling laws. The latter results agree with a theoretical prediction from a stochastic model, and also with an exact result for the scaling exponent we derived for the specific class of models investigated. We discuss the consequences of the scaling laws in terms of the predictability of a final state (extinction of either species) in an ecological experiment.Comment: 24 pages (preprint format), 6 figure

Perturbações em sistemas com variabilidade da dimensão instável transversal

A variabilidade da dimensão instável (VDI) é uma forma extrema de não-hiperbolicidade. É um fenômeno estruturalmente estável, típico para sistemas caóticos de alta dimensionalidade, que implica restrições severas ao sombreamento de soluções perturbadas. As perturbações s são inevitáveis na modelagem de fenômenos fíısicos, uma vez que nenhum sistema pode ser isolado completamente, os estados e os parâmetros não podem ser determinados sem incertezas e qualquer abordagem numérica dos modelos é afetada por erros de arredondamento e/ou truncamento. Portanto, a falta da sombreabilidade em sistemas exibindo VDI apresenta um desafio à modelagem. Visando revelar os efeitos das perturbações, uma classe desses sistemas não hiperbó licos é estudada. Esses sistemas apresentam variabilidade da dimensão instável transversal (VDIT), significando que a dinâmica pode ser decomposta na forma de um produto direto assimétrico, i. e. o espação de fase é dividido em dois componentes: um hiperbólico e caótico, dito longitudinal, e um transversal e não-hiperbólico. Mais ainda, na ausência de perturbações, o componente longitudinal é um atrator global do sistema. Um protótipo composto de dois mapas caoticos lineares por partes acoplados é apresentado para o estudo dos efeitos da VDIT. Esse sistema possui um subespaço invariante S que caracteriza a sincronização completa de caos e a VDI, quando presente, é transversal a esse subespaço. Valendo-se da linearidade (por partes) das equações, um método analítico para o cálculo das órbitas periódicas instáveis é apresentado. O conjunto de todas as órbitas periódicas instáveis (OPIs) é um dos fundamentos da dinâmica caótica e suas propriedades fornecem informaões, valiosas sobre o comportamento assintótico do sistema como, por exemplo, a medida natural invariante. Assim, a intensidade da VDIT é estudada analiticamente pelo cálculo da medida de contraste, que quantifica a diferença entre o peso estatístico associado às OPIs com dimensão instável distintas. O efeito das perturbações é modelado pela introdução de um pequeno desvio nos parâmetros, ao invés da adição de ruído, a fim de manter o determinismo do modelo. Consequentemente, a caracterização da dinâmica em termos das OPIs ainda é possível. Demonstra-se a existência de um conjunto denso G de OPIs fora do subespaço invariante consistente com um repulsor caótico. Na presença de perturbações, G se funde com o conjunto H das OPIs previamente em S, dando origem a um novo estado estacionario não-hiperbólico. A análise de G ∪H fornece uma explicação topológica ao comportamento de sistemas com variabilidade da dimensão instável sob a açãoo de perturbações. Mais ainda, a relação entre o conjunto de OPIs imersas em um atrator caótico e sua medida natural, provada apenas para sistemas hiperbólicos, é aplicada com sucesso nesse sistema: mostra-se que o erro entre as medidas naturais estimadas numericamente e pelas OPIs é decrescente com p, o período das OPIs consideradas. Conjectura-se, portanto, a coincidência entre ambas no limite . Logo, apresenta-se uma resposta positiva à estimativa numérica da medida natural em sistemas não-hiperbólicos via variabilidade da dimensão instável.Unstable dimension variability (UDV) is an extreme form of nonhyperbolicity. It is a structurally stable phenomenon, typical for high dimensional chaotic systems, which implies severe restrictions to shadowing of perturbed solutions. Perturbations are unavoidable in modelling Physical phenomena, since no system can be made completely isolated, states and parameters cannot be determined without uncertainties and any numeric approach to such models is affected by truncation and/or roundoff errors. Thus, the lack of shadowability in systems exhibiting UDV presents a challenge for modelling. Aiming to unveil the effect of perturbations a class of nonhyperbolic systems is studied. These systems present transversal unstable dimension variability (TUDV), which means the dynamics can be split in a skew direct product form, i. e. the phase space is decomposed in two components: a hyperbolic chaotic one, called longitudinal, and a nonhyperbolic transversal one. Moreover, in the absence of perturbations, the longitudinal component is a global attractor of the system. A prototype composed of two coupled piecewise-linear chaotic maps is presented in order to study the TUDV effects. This system has an invariant subspace S which characterizes the complete chaos synchronization and UDV, when present, is transversal to it. Taking advantage of (piecewise) linearity of the equations, an analytical method for unstable periodic orbits computation is presented. The set of all unstable periodic orbits (UPOs) is one of the building block of chaotic dynamics and its properties provide valuable informations about the asymptotic behaviour of the system as, for instance, the invariant natural measure. Therefore, the TUDVs intensity is analytically studied by computing the contrast measure, which quantifies the difference between the statistical weights associated to UPOs with different unstable dimension. The effect of perturbations is modelled by the introduction of a small parameter mismatch, instead of noise addition, in order to keep the models determinism. Consequently, the characterization of dynamics by means of UPOs is still possible. It is shown the existence of a dense set G of UPOs outside the invariant subspace consistent with a chaotic repeller. When perturbation takes place, G merges with the set H of UPOs previously in S, given rise to a new nonhyperbolic stationary state. The analysis of G ∪H provides a topological explanation to the behaviour of systems with TUDV under perturbations. Moreover, the relation between the set of UPOs embedded in a chaotic attractor and its natural measure, proven only for hyperbolic systems, is successfully applied to this system: the error between the natural measure estimated both numerically and by means of UPOs is shown to be decreasing with p, the considered UPOs period. It is conjectured the coincidence between both in limit. Hence, a positive answer to reliability of numerical estimation to natural measure in nonhyperbolic systems via unstable dimension variability is presented

NEOTROPICAL FRESHWATER FISHES: A dataset of occurrence and abundance of freshwater fishes in the Neotropics

The Neotropical region hosts 4225 freshwater fish species, ranking first among the world's most diverse regions for freshwater fishes. Our NEOTROPICAL FRESHWATER FISHES data set is the first to produce a large-scale Neotropical freshwater fish inventory, covering the entire Neotropical region from Mexico and the Caribbean in the north to the southern limits in Argentina, Paraguay, Chile, and Uruguay. We compiled 185,787 distribution records, with unique georeferenced coordinates, for the 4225 species, represented by occurrence and abundance data. The number of species for the most numerous orders are as follows: Characiformes (1289), Siluriformes (1384), Cichliformes (354), Cyprinodontiformes (245), and Gymnotiformes (135). The most recorded species was the characid Astyanax fasciatus (4696 records). We registered 116,802 distribution records for native species, compared to 1802 distribution records for nonnative species. The main aim of the NEOTROPICAL FRESHWATER FISHES data set was to make these occurrence and abundance data accessible for international researchers to develop ecological and macroecological studies, from local to regional scales, with focal fish species, families, or orders. We anticipate that the NEOTROPICAL FRESHWATER FISHES data set will be valuable for studies on a wide range of ecological processes, such as trophic cascades, fishery pressure, the effects of habitat loss and fragmentation, and the impacts of species invasion and climate change. There are no copyright restrictions on the data, and please cite this data paper when using the data in publications

Neotropical freshwater fisheries : A dataset of occurrence and abundance of freshwater fishes in the Neotropics

The Neotropical region hosts 4225 freshwater fish species, ranking first among the world's most diverse regions for freshwater fishes. Our NEOTROPICAL FRESHWATER FISHES data set is the first to produce a large-scale Neotropical freshwater fish inventory, covering the entire Neotropical region from Mexico and the Caribbean in the north to the southern limits in Argentina, Paraguay, Chile, and Uruguay. We compiled 185,787 distribution records, with unique georeferenced coordinates, for the 4225 species, represented by occurrence and abundance data. The number of species for the most numerous orders are as follows: Characiformes (1289), Siluriformes (1384), Cichliformes (354), Cyprinodontiformes (245), and Gymnotiformes (135). The most recorded species was the characid Astyanax fasciatus (4696 records). We registered 116,802 distribution records for native species, compared to 1802 distribution records for nonnative species. The main aim of the NEOTROPICAL FRESHWATER FISHES data set was to make these occurrence and abundance data accessible for international researchers to develop ecological and macroecological studies, from local to regional scales, with focal fish species, families, or orders. We anticipate that the NEOTROPICAL FRESHWATER FISHES data set will be valuable for studies on a wide range of ecological processes, such as trophic cascades, fishery pressure, the effects of habitat loss and fragmentation, and the impacts of species invasion and climate change. There are no copyright restrictions on the data, and please cite this data paper when using the data in publications

NEOTROPICAL FRESHWATER FISHES: A dataset of occurrence and abundance of freshwater fishes in the Neotropics

The Neotropical region hosts 4225 freshwater fish species, ranking first among the world's most diverse regions for freshwater fishes. Our NEOTROPICAL FRESHWATER FISHES data set is the first to produce a large-scale Neotropical freshwater fish inventory, covering the entire Neotropical region from Mexico and the Caribbean in the north to the southern limits in Argentina, Paraguay, Chile, and Uruguay. We compiled 185,787 distribution records, with unique georeferenced coordinates, for the 4225 species, represented by occurrence and abundance data. The number of species for the most numerous orders are as follows: Characiformes (1289), Siluriformes (1384), Cichliformes (354), Cyprinodontiformes (245), and Gymnotiformes (135). The most recorded species was the characid Astyanax fasciatus (4696 records). We registered 116,802 distribution records for native species, compared to 1802 distribution records for nonnative species. The main aim of the NEOTROPICAL FRESHWATER FISHES data set was to make these occurrence and abundance data accessible for international researchers to develop ecological and macroecological studies, from local to regional scales, with focal fish species, families, or orders. We anticipate that the NEOTROPICAL FRESHWATER FISHES data set will be valuable for studies on a wide range of ecological processes, such as trophic cascades, fishery pressure, the effects of habitat loss and fragmentation, and the impacts of species invasion and climate change. There are no copyright restrictions on the data, and please cite this data paper when using the data in publications

NEOTROPICAL FRESHWATER FISHES: A dataset of occurrence and abundance of freshwater fishes in the Neotropics

The Neotropical region hosts 4225 freshwater fish species, ranking first among the world's most diverse regions for freshwater fishes. Our NEOTROPICAL FRESHWATER FISHES data set is the first to produce a large-scale Neotropical freshwater fish inventory, covering the entire Neotropical region from Mexico and the Caribbean in the north to the southern limits in Argentina, Paraguay, Chile, and Uruguay. We compiled 185,787 distribution records, with unique georeferenced coordinates, for the 4225 species, represented by occurrence and abundance data. The number of species for the most numerous orders are as follows: Characiformes (1289), Siluriformes (1384), Cichliformes (354), Cyprinodontiformes (245), and Gymnotiformes (135). The most recorded species was the characid Astyanax fasciatus (4696 records). We registered 116,802 distribution records for native species, compared to 1802 distribution records for nonnative species. The main aim of the NEOTROPICAL FRESHWATER FISHES data set was to make these occurrence and abundance data accessible for international researchers to develop ecological and macroecological studies, from local to regional scales, with focal fish species, families, or orders. We anticipate that the NEOTROPICAL FRESHWATER FISHES data set will be valuable for studies on a wide range of ecological processes, such as trophic cascades, fishery pressure, the effects of habitat loss and fragmentation, and the impacts of species invasion and climate change. There are no copyright restrictions on the data, and please cite this data paper when using the data in publications