Fractal spatialities

This paper argues for the use of ‘fractals’ in theorising sociospatial relations. From a realist position, a nonmathematical but nonmetaphoric and descriptive view of ‘fractals’ is advanced. Insights from the natural sciences are combined with insights on the position of the observer from Luhmann and notions of assemblages and repetitions from Deleuze. It is argued that the notion of ‘fractals’ can augment current understanding of sociospatialities in three ways. First, it can pose questions about the scalar position of the observer or the grain of observation; second, as a signifier of particular attributes, it prompts observation and description of particular structuring processes; and third, the epistemic access afforded by the concept can open up possibilities for transformative interventions and thereby inform the same. The theoretical usefulness of the concept is demonstrated by discussing the territory, place, scale, and networks (TPSN) model for theorising sociospatial relations advanced by B Jessop, N Brenner, and M Jones in their 2008 paper “Theorizing sociospatial relations”, published in this journal (volume 26, pages 389–401). It is suggested that a heuristic arising from a ‘fractal’ ontology can contribute to a polymorphous, as opposed to polyvalent, understanding of sociospatial relations.

Fractal spaces in planning and governance

This paper discusses concepts of space within the planning literature, the issues they give rise to and the gaps they reveal. It then introduces the notion of 'fractals' borrowed from complexity theory and illustrates how it unconsciously appears in planning practice. It then moves on to abstract the core dynamics through which fractals can be consciously applied and illustrates their working through a reinterpretation of the People's Planning Campaign of Kerala, India. Finally it highlights the key contribution of the fractal concept and the advantages that this conceptualisation brings to planning.

Consistency in approximate entropy given by a volumetric estimate

Non-linear methods for estimating variability in time-series are currently of widespread use. Among such methods are approximate entropy (ApEn) and sample approximate entropy (SampEn). The applicability of ApEn and SampEn in analyzing data is evident and their use is increasing. However, consistency is a point of concern in these tools, i.e., the classification of the temporal organization of a data set might indicate a relative less ordered series in relation to another when the opposite is true. As highlighted by their proponents themselves, ApEn and SampEn might present incorrect results due to this lack of consistency. In this study, we present a method which gains consistency by using ApEn repeatedly in a wide range of combinations of window lengths and matching error tolerance. The tool is called volumetric approximate entropy, vApEn. We analyze nine artificially generated prototypical time-series with different degrees of temporal order (combinations of sine waves, logistic maps with different control parameter values, random noises). While ApEn/SampEn clearly fail to consistently identify the temporal order of the sequences, vApEn correctly do. In order to validate the tool we performed shuffled and surrogate data analysis. Statistical analysis confirmed the consistency of the method. (C) 2008 Elsevier Ltd. All rights reserved.

TRANSITIVE CIRCLE EXCHANGE TRANSFORMATIONS WITH FLIPS

We study the existence of transit we exchange transformations with flips defined on the unit circle S(1). We provide a complete answer to the question of whether there exists a transitive exchange transformation of S(1) defined on a subintervals and having f flips.

Clustering and diffusion in a symplectic map lattice with non-local coupling

We study a symplectic chain with a non-local form of coupling by means of a standard map lattice where the interaction strength decreases with the lattice distance as a power-law, in Such a way that one can pass continuously from a local (nearest-neighbor) to a global (mean-field) type of coupling. We investigate the formation of map clusters, or spatially coherent structures generated by the system dynamics. Such clusters are found to be related to stickiness of chaotic phase-space trajectories near periodic island remnants, and also to the behavior of the diffusion coefficient. An approximate two-dimensional map is derived to explain some of the features of this connection. (C) 2008 Elsevier Ltd. All rights reserved.

A scenario for torus T(2) destruction via a global bifurcation

We show a scenario of a two-frequeney torus breakdown, in which a global bifurcation occurs due to the collision of a quasi-periodic torus T(2) with saddle points, creating a heteroclinic saddle connection. We analyze the geometry of this torus-saddle collision by showing the local dynamics and the invariant manifolds (global dynamics) of the saddle points. Moreover, we present detailed evidences of a heteroclinic saddle-focus orbit responsible for the type-if intermittency induced by this global bifurcation. We also characterize this transition to chaos by measuring the Lyapunov exponents and the scaling laws. (C) 2007 Elsevier Ltd. All rights reserved.

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing. (C) 2007 Elsevier Ltd. All rights reserved.

Experimental identification of chaotic fibers

In a 2D parameter space, by using nine experimental time series of a Clitia`s circuit, we characterized three codimension-1 chaotic fibers parallel to a period-3 window. To show the local preservation of the properties of the chaotic attractors in each fiber, we applied the closed return technique and two distinct topological methods. With the first topological method we calculated the linking, numbers in the sets of unstable periodic orbits, and with the second one we obtained the symbolic planes and the topological entropies by applying symbolic dynamic analysis. (C) 2007 Elsevier Ltd. All rights reserved.

Velocity-dependent KAM islands

We revisit the non-dissipative time-dependent annular billiard and we consider the chaotic dynamics in two planes of conjugate variables in order to describe the behavior of the growth, or saturation, of the mean velocity of an ensemble of particles. We observed that the changes in the 4-d phase space occur without changing any parameter. They occur depending on where the initial conditions start. The emerging KAM islands interfere in the behavior of the particle dynamics especially in the Fermi acceleration mechanism. We show that Fermi acceleration can be suppressed, without dissipation, even considering the non-dissipative energy context. (C) 2011 Published by Elsevier Ltd.

Financial time series analysis : Chaos and neurodynamics approach

This work aims at combining the Chaos theory postulates and Artificial Neural Networks classification and predictive capability, in the field of financial time series prediction. Chaos theory, provides valuable qualitative and quantitative tools to decide on the predictability of a chaotic system. Quantitative measurements based on Chaos theory, are used, to decide a-priori whether a time series, or a portion of a time series is predictable, while Chaos theory based qualitative tools are used to provide further observations and analysis on the predictability, in cases where measurements provide negative answers. Phase space reconstruction is achieved by time delay embedding resulting in multiple embedded vectors. The cognitive approach suggested, is inspired by the capability of some chartists to predict the direction of an index by looking at the price time series. Thus, in this work, the calculation of the embedding dimension and the separation, in Takens‘ embedding theorem for phase space reconstruction, is not limited to False Nearest Neighbor, Differential Entropy or other specific method, rather, this work is interested in all embedding dimensions and separations that are regarded as different ways of looking at a time series by different chartists, based on their expectations. Prior to the prediction, the embedded vectors of the phase space are classified with Fuzzy-ART, then, for each class a back propagation Neural Network is trained to predict the last element of each vector, whereas all previous elements of a vector are used as features.

Análise fractal da vasculatura arteriolar e venular da retina em pessoas sem doença ocular

Although it has been suggested that retinal vasculature is a diffusion-limited aggregation (DLA) fractal, no study has been dedicated to standardizing its fractal analysis . The aims of this project was to standardize a method to estimate the fractal dimensions of retinal vasculature and to characterize their normal values; to determine if this estimation is dependent on skeletization and on segmentation and calculation methods; to assess the suitability of the DLA model and to determine the usefulness of log-log graphs in characterizing vasculature fractality . To achieve these aims, the information, mass-radius and box counting dimensions of 20 eyes vasculatures were compared when the vessels were manually or computationally segmented; the fractal dimensions of the vasculatures of 60 eyes of healthy volunteers were compared with those of 40 DLA models and the log-log graphs obtained were compared with those of known fractals and those of non-fractals. The main results were: the fractal dimensions of vascular trees were dependent on segmentation methods and dimension calculation methods, but there was no difference between manual segmentation and scale-space, multithreshold and wavelet computational methods; the means of the information and box dimensions for arteriolar trees were 1.29. against 1.34 and 1.35 for the venular trees; the dimension for the DLA models were higher than that for vessels; the log-log graphs were straight, but with varying local slopes, both for vascular trees and for fractals and non-fractals. This results leads to the following conclusions: the estimation of the fractal dimensions for retinal vasculature is dependent on its skeletization and on the segmentation and calculation methods; log-log graphs are not suitable as a fractality test; the means of the information and box counting dimensions for the normal eyes were 1.47 and 1.43, respectively, and the DLA model with optic disc seeding is not sufficient for retinal vascularization modeling

Fragmentos do pensamento dialético na história da construção das ciências da natureza

Este trabalho tem por objetivo identificar uma possível inclinação das ciências naturais em direção ao materialismo dialético. Para tanto, procura-se apresentar a história da dialética a partir da discussão racionalismo/empirismo moderno e seus desdobramentos até as tendências dialéticos contemporâneas. Os autores discutidos são Kant, Hegel, Marx, Engels, Lenin, Horkheimer, Marcuse, Habermas, Bachelard e suas escolas epistemológicas, completadas por Althusser, Lefebvre e Kedrov. Ao lado desses autores discutem-se outros, das duas últimas décadas, procurando extrair-lhes o olhar dialético, oculto em seus discursos acerca da ciência do fim do século. Também se procura encontrar na mecânica quântica, nos fractais, na lógica para-consistente, nos modelos matemáticos e na biologia antideterminista, argumentos para existência de uma forma de abordagem dialética da natureza. Por último, procura-se refletir acerca dos motivos da resistência ao método dialético apresentado pela maioria dos cientistas ocidentais e, sua possível superação.

Estudo do acoplamento entre superfícies seletivas de frequência assimétricas em estruturas de multicamadas

This work presents the development of new microwaves structures, filters and high gain antenna, through the cascading of frequency selective surfaces, which uses fractals Dürer and Minkowski patches as elements, addition of an element obtained from the combination of the other two simple the cross dipole and the square spiral. Frequency selective surfaces (FSS) includes a large area of Telecommunications and have been widely used due to its low cost, low weight and ability to integrate with others microwaves circuits. They re especially important in several applications, such as airplane, antennas systems, radomes, rockets, missiles, etc. FSS applications in high frequency ranges have been investigated, as well as applications of cascading structures or multi-layer, and active FSS. In this work, we present results for simulated and measured transmission characteristics of cascaded structures (multilayer), aiming to investigate the behavior of the operation in terms of bandwidth, one of the major problems presented by frequency selective surfaces. Comparisons are made with simulated results, obtained using commercial software such as Ansoft DesignerTM v3 and measured results in the laboratory. Finally, some suggestions are presented for future works on this subject

Antenas de microfita com patch quase-fractal para aplicações em Redes WPAN/WLAN

The microstrip antennas are in constant evidence in current researches due to several advantages that it presents. Fractal geometry coupled with good performance and convenience of the planar structures are an excellent combination for design and analysis of structures with ever smaller features and multi-resonant and broadband. This geometry has been applied in such patch microstrip antennas to reduce its size and highlight its multi-band behavior. Compared with the conventional microstrip antennas, the quasifractal patch antennas have lower frequencies of resonance, enabling the manufacture of more compact antennas. The aim of this work is the design of quasi-fractal patch antennas through the use of Koch and Minkowski fractal curves applied to radiating and nonradiating antenna s edges of conventional rectangular patch fed by microstrip inset-fed line, initially designed for the frequency of 2.45 GHz. The inset-fed technique is investigated for the impedance matching of fractal antennas, which are fed through lines of microstrip. The efficiency of this technique is investigated experimentally and compared with simulations carried out by commercial software Ansoft Designer used for precise analysis of the electromagnetic behavior of antennas by the method of moments and the neural model proposed. In this dissertation a study of literature on theory of microstrip antennas is done, the same study is performed on the fractal geometry, giving more emphasis to its various forms, techniques for generation of fractals and its applicability. This work also presents a study on artificial neural networks, showing the types/architecture of networks used and their characteristics as well as the training algorithms that were used for their implementation. The equations of settings of the parameters for networks used in this study were derived from the gradient method. It will also be carried out research with emphasis on miniaturization of the proposed new structures, showing how an antenna designed with contours fractals is capable of a miniaturized antenna conventional rectangular patch. The study also consists of a modeling through artificial neural networks of the various parameters of the electromagnetic near-fractal antennas. The presented results demonstrate the excellent capacity of modeling techniques for neural microstrip antennas and all algorithms used in this work in achieving the proposed models were implemented in commercial software simulation of Matlab 7. In order to validate the results, several prototypes of antennas were built, measured on a vector network analyzer and simulated in software for comparison

Análise e projeto de superfícies seletivas de frequência com elementos pré-fractais para aplicações em comunicações indoor

In this thesis, a frequency selective surface (FSS) consists of a two-dimensional periodic structure mounted on a dielectric substrate, which is capable of selecting signals in one or more frequency bands of interest. In search of better performance, more compact dimensions, low cost manufacturing, among other characteristics, these periodic structures have been continually optimized over time. Due to its spectral characteristics, which are similar to band-stop or band-pass filters, the FSSs have been studied and used in several applications for more than four decades. The design of an FSS with a periodic structure composed by pre-fractal elements facilitates the tuning of these spatial filters and the adjustment of its electromagnetic parameters, enabling a compact design which generally has a stable frequency response and superior performance relative to its euclidean counterpart. The unique properties of geometric fractals have shown to be useful, mainly in the production of antennas and frequency selective surfaces, enabling innovative solutions and commercial applications in microwave range. In recent applications, the FSSs modify the indoor propagation environments (emerging concept called wireless building ). In this context, the use of pre-fractal elements has also shown promising results, allowing a more effective filtering of more than one frequency band with a single-layer structure. This thesis approaches the design of FSSs using pre-fractal elements based on Vicsek, Peano and teragons geometries, which act as band-stop spatial filters. The transmission properties of the periodic surfaces are analyzed to design compact and efficient devices with stable frequency responses, applicable to microwave frequency range and suitable for use in indoor communications. The results are discussed in terms of the electromagnetic effect resulting from the variation of parameters such as: fractal iteration number (or fractal level), scale factor, fractal dimension and periodicity of FSS, according the pre-fractal element applied on the surface. The analysis of the fractal dimension s influence on the resonant properties of a FSS is a new contribution in relation to researches about microwave devices that use fractal geometry. Due to its own characteristics and the geometric shape of the Peano pre-fractal elements, the reconfiguration possibility of these structures is also investigated and discussed. This thesis also approaches, the construction of efficient selective filters with new configurations of teragons pre-fractal patches, proposed to control the WLAN coverage in indoor environments by rejecting the signals in the bands of 2.4~2.5 GHz (IEEE 802.11 b) and 5.0~6.0 GHz (IEEE 802.11a). The FSSs are initially analyzed through simulations performed by commercial software s: Ansoft DesignerTM and HFSSTM. The fractal design methodology is validated by experimental characterization of the built prototypes, using alternatively, different measurement setups, with commercial horn antennas and microstrip monopoles fabricated for low cost measurements