Fractional Order Junctions

Gottfried Leibniz generalized the derivation and integration, extending the operators from integer up to real, or even complex, orders. It is presently recognized that the resulting models capture long term memory effects difficult to describe by classical tools. Leon Chua generalized the set of lumped electrical elements that provide the building blocks in mathematical models. His proposal of the memristor and of higher order elements broadened the scope of variables and relationships embedded in the development of models. This paper follows the two directions and proposes a new logical step, by generalizing the concept of junction. Classical junctions interconnect system elements using simple algebraic restrictions. Nevertheless, this simplistic approach may be misleading in the presence of unexpected dynamical phenomena and requires including additional “parasitic” elements. The novel γ-junction includes, as special cases, the standard series and parallel connections and allows a new degree of freedom when building models. The proposal motivates the search for experimental and real world manifestations of the abstract conjectures.

The Persistence of Memory

This paper analyzes several natural and man-made complex phenomena in the perspective of dynamical systems. Such phenomena are often characterized by the absence of a characteristic length-scale, long range correlations and persistent memory, which are features also associated to fractional order systems. For each system, the output, interpreted as a manifestation of the system dynamics, is analyzed by means of the Fourier transform. The amplitude spectrum is approximated by a power law function and the parameters are interpreted as an underlying signature of the system dynamics. The complex systems under analysis are then compared in a global perspective in order to unveil and visualize hidden relationships among them.

Fractional Order Description of DNA

This study addresses the deoxyribonucleic acid (DNA) and proposes a procedure based on the association of statistics, information theory, signal processing, Fourier analysis and fractional calculus for describing fundamental characteristics of the DNA. In a first phase the 24 chromosomes of the Human are evaluated. In a second phase, 10 chromosomes for different species are also processed and the results compared. The results reveal invariance in the description and close resemblances with fractional Brownian motion.

Discrete fractional order system vibrations

A theory of free vibrations of discrete fractional order (FO) systems with a finite number of degrees of freedom (dof) is developed. A FO system with a finite number of dof is defined by means of three matrices: mass inertia, system rigidity and FO elements. By adopting a matrix formulation, a mathematical description of FO discrete system free vibrations is determined in the form of coupled fractional order differential equations (FODE). The corresponding solutions in analytical form, for the special case of the matrix of FO properties elements, are determined and expressed as a polynomial series along time. For the eigen characteristic numbers, the system eigen main coordinates and the independent eigen FO modes are determined. A generalized function of visoelastic creep FO dissipation of energy and generalized forces of system with no ideal visoelastic creep FO dissipation of energy for generalized coordinates are formulated. Extended Lagrange FODE of second kind, for FO system dynamics, are also introduced. Two examples of FO chain systems are analyzed and the corresponding eigen characteristic numbers determined. It is shown that the oscillatory phenomena of a FO mechanical chain have analogies to electrical FO circuits. A FO electrical resistor is introduced and its constitutive voltage–current is formulated. Also a function of thermal energy FO dissipation of a FO electrical relation is discussed.

State Space Analysis of Forest Fires

This paper studies forest fires from the perspective of dynamical systems. Burnt area, precipitation and atmospheric temperatures are interpreted as state variables of a complex system and the correlations between them are investigated by means of different mathematical tools. First, we use mutual information to reveal potential relationships in the data. Second, we adopt the state space portrait to characterize the system’s behavior. Third, we compare the annual state space curves and we apply clustering and visualization tools to unveil long-range patterns. We use forest fire data for Portugal, covering the years 1980–2003. The territory is divided into two regions (North and South), characterized by different climates and vegetation. The adopted methodology represents a new viewpoint in the context of forest fires, shedding light on a complex phenomenon that needs to be better understood in order to mitigate its devastating consequences, at both economical and environmental levels.

An Extension of Estimation of Domain of Attraction for Fractional Order Linear System Subject to Saturation Control

This paper employs the Lyapunov direct method for the stability analysis of fractional order linear systems subject to input saturation. A new stability condition based on saturation function is adopted for estimating the domain of attraction via ellipsoid approach. To further improve this estimation, the auxiliary feedback is also supported by the concept of stability region. The advantages of the proposed method are twofold: (1) it is straightforward to handle the problem both in analysis and design because of using Lyapunov method, (2) the estimation leads to less conservative results. A numerical example illustrates the feasibility of the proposed method.

Modeling Vegetable Fractals by means of Fractional-order Equations

This paper characterizes four ‘fractal vegetables’: (i) cauliflower (brassica oleracea var. Botrytis); (ii) broccoli (brassica oleracea var. italica); (iii) round cabbage (brassica oleracea var. capitata) and (iv) Brussels sprout (brassica oleracea var. gemmifera), by means of electrical impedance spectroscopy and fractional calculus tools. Experimental data is approximated using fractional-order models and the corresponding parameters are determined with a genetic algorithm. The Havriliak-Negami five-parameter model fits well into the data, demonstrating that classical formulae can constitute simple and reliable models to characterize biological structures.

Integer and Fractional Order Entropy Analysis of Earthquake Data-series

This paper studies the statistical distributions of worldwide earthquakes from year 1963 up to year 2012. A Cartesian grid, dividing Earth into geographic regions, is considered. Entropy and the Jensen–Shannon divergence are used to analyze and compare real-world data. Hierarchical clustering and multi-dimensional scaling techniques are adopted for data visualization. Entropy-based indices have the advantage of leading to a single parameter expressing the relationships between the seismic data. Classical and generalized (fractional) entropy and Jensen–Shannon divergence are tested. The generalized measures lead to a clear identification of patterns embedded in the data and contribute to better understand earthquake distributions.

Barnacle Allergy: Allergen Characterization and Cross-Reactivity with Mites

Background: Barnacles are a type of seafood with worldwide distribution and abundant along the shores of temperate seas. They are particularly appreciated and regularly consumed in Portugal as well as in Spain, France and South America, but barnacle allergy is a rare condition of which there is only one reference in the indexed literature. The molecular allergens and possible cross-reactivity phenomena implicated (namely with mites) have not been established. Objective: To demonstrate the IgE-mediated allergy to barnacle and to identify the proteins implicated as well as possible cross-reactivity phenomena with mites. Methods: We report the clinical and laboratory data of five patients with documented IgE-mediated allergy to barnacle. The diagnosis was based on a suggestive clinical history combined with positive skin prick tests (SPT) to barnacle – prick to prick method. Two barnacle extracts were prepared (raw and cooked barnacle) and sodium dodecylsulphate polyacrylamide gel electrophoresis (SDS-PAGE) and IgE-immunoblotting were performed. An immunoblotting inhibition assay with Dermatophagoides pteronyssinus was also done in order to evaluate cross-reactivity. Results: All patients had mite-related asthma and the allergic rhinoconjunctivitis; they all experienced mucocutaneous symptoms. All of them had positive SPT to barnacle, and the immunoblotting showed several allergenic fractions with a wide molecular weight range (19 – 94 kDa). The D. pteronyssinus extract inhibited several IgE-binding protein fractions in the barnacle extract. Conclusions: We describe five patients with IgE-mediated barnacle allergy. We also describe a group of IgEbinding+ proteins between 30 and 75 kDa as the allergenic fractions of this type of Crustacea. Cross-reactivity with D. pteronyssinus was demonstrated in two cases.

Coordinated Maneuver for Gradient Search Using Multiple AUV's

OCEANS 2003. Proceedings (Volume:1 )

Serpentine bacteria influence metal translocation and bioconcentration of Brassica juncea and Ricinus communis grown in multi-metal polluted soils

The aim of this study was to assess the effects of inoculation of rhizosphere or endophytic bacteria (Psychrobacter sp. SRS8 and Pseudomonas sp. A3R3, respectively) isolated from a serpentine environment on the plant growth and the translocation and accumulation of Ni, Zn, and Fe by Brassica juncea and Ricinus communis on a multi-metal polluted serpentine soil (SS). Field collected SS was diluted to 0, 25, 50, and 75% with pristine soil in order to obtain a range of heavy metal concentrations and used in microcosm experiments. Regardless of inoculation with bacteria, the biomass of both plant species decreased with increase of the proportion of SS. Inoculation of plants with bacteria significantly increased the plant biomass and the heavy metal accumulation compared with non-inoculated control in the presence of different proportion of SS, which was attributed to the production of plant growth promoting and/or metal mobilizing metabolites by bacteria. However, SRS8 showed a maximum increase in the biomass of the test plants grown even in the treatment of 75% SS. In turn, A3R3 showed maximum effects on the accumulation of heavy metals in both plants. Regardless of inoculation of bacteria and proportion of SS, both plant species exhibited low values of bioconcentration factor (<1) for Ni and Fe. The inoculation of both bacterial strains significantly increased the translocation factor (TF) of Ni while decreasing the TF of Zn in both plant species. Besides this contrasting effect, the TFs of all metals were <1, indicating that all studied bacteria–plant combinations are suitable for phytostabilization. This study demonstrates that the bacterial isolates A3R3 and SRS8 improved the growth of B. juncea and R. communis in SS soils and have a great potential to be used as inoculants in phytostabilization scenarios of multi-metal contaminated soils.

Gender differences in internet usage habits: a case study in higher education

The usage of information and communication technologies has been growing among students and teachers. In order to improve the use of the Internet as a tool to support teaching and learning it is necessary to understand the Internet usage habits of students. Thus, a study was conducted with 1397 students from five schools of the Polytechnic of Porto. The data was collected through an online questionnaire abd was analized by age range, gender and scientific field. In this paper,gender differences are analyzed and presented in 3 dimensions: type of Internet usage, communication tools and the role of the Internet tools in education.

Control and Dynamics of Fractional Order Systems

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades due to the progress in the area of nonlinear dynamics. This article discusses several applications of fractional calculus in science and engineering, namely: the control of heat systems, the tuning of PID controllers based on fractional calculus concepts and the dynamics in hexapod locomotion.

Design of Digital Fractional-Order Integrators and Differentiators by Least Squares

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Improved Numerical Simulation for a Novel Adaptive Control Using Fractional Order Derivatives

A novel control technique is investigated in the adaptive control of a typical paradigm, an approximately and partially modeled cart plus double pendulum system. In contrast to the traditional approaches that try to build up ”complete” and ”permanent” system models it develops ”temporal” and ”partial” ones that are valid only in the actual dynamic environment of the system, that is only within some ”spatio-temporal vicinity” of the actual observations. This technique was investigated for various physical systems via ”preliminary” simulations integrating by the simplest 1st order finite element approach for the time domain. In 2004 INRIA issued its SCILAB 3.0 and its improved numerical simulation tool ”Scicos” making it possible to generate ”professional”, ”convenient”, and accurate simulations. The basic principles of the adaptive control, the typical tools available in Scicos, and others developed by the authors, as well as the improved simulation results and conclusions are presented in the contribution.