Monthly newspaper for Greenhills, Ohio. August, 2014
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T-Feigenbaum Graphs Ghj - Free download as PDF File (.pdf), Text File (.txt) or read online for free. We analyze the properties of networks obtained from the trajectories of unimodal maps at the transition to chaos via the horizontal visibility (HV) algorithm. We find that the network degrees fluctuate at all scales with amplitude that increases as the size of the network grows, and can be described by a spectrum of graph-theoretical generalized Lyapunov exponents. We further define an entropy growth rate that describes the amount of information created along paths in network space, and find that such entropy growth rate coincides with the spectrum of generalized graph-theoretical exponents, constituting a set of Pesin-like identities for the network.The horizontal visibility (HV) algorithm is a general method to convert time series data into a graph [8,9] and is concisely stated as follows: assign a node i to each datum xi of the time series {xi}i=1,...,N of N real data, and then connect any pair of nodes i, j if their associated data fulfill the criterion xi , x j > xn for all n such that i < n < j. The capability of the method as well as other variants to transfer properties of different types of time series into their resultant graphs has been demonstrated in recent works [10,11]. When the series under study are the trajectories within the attractors generated by unimodal or circle maps the application of the HV algorithm yield subfamilies of visibility graphs, named Feigenbaum and quasiperiodic graphs respectively, that render the known low-dimensional routes to chaos in a new setting [6,7,12]. For illustrative purposes, in Fig. 1 we show a hierarchy of Feigenbaum graphs obtained along the period-doubling bifurcation cascade of unimodal maps. At the accumulation point of the cascade the trajectories within its non-chaotic multifractal attractor become aperiodic, and in analogy with the original Feigenbaum treatment [13], the associated Feigenbaum graphs evidence scaling properties that can be exploited by an appropriate graph-theoretical Renormalization Group transformation [6,7]. Because the order of visits of positions of periodic attractors or of bands of chaotic attractors in unimodal maps are universal, a relevant consequence of the HV criterion is that the resulting Feigenbaum graph at the onset of chaos is the same for every unimodal map. That is, it is independent of the shape and nonlinearity of the map [6,7]. This permits us to concentrate our study on a specific unimodal map, that for simplicity will be the logistic map, and claim generality of the results.
Subtítulo: LEARNERS Origem: IMPORTADO Editora: MCGRAW-HILL PROFESSI Coleção: VERBS AND ESSENTIALS OF GRAMMAR SERIES Assunto: ENSINO DE LÍNGUAS I...
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