Fractional Dynamics in Forest Fires

Every year forest fires consume large areas, being a major concern in many countries like Australia, United States and Mediterranean Basin European Countries (e.g., Portugal, Spain, Italy and Greece). Understanding patterns of such events, in terms of size and spatiotemporal distributions, may help to take measures beforehand in view of possible hazards and decide strategies of fire prevention, detection and suppression. Traditional statistical tools have been used to study forest fires. Nevertheless, those tools might not be able to capture the main features of fires complex dynamics and to model fire behaviour [1]. Forest fires size-frequency distributions unveil long range correlations and long memory characteristics, which are typical of fractional order systems [2]. Those complex correlations are characterized by self-similarity and absence of characteristic length-scale, meaning that forest fires exhibit power-law (PL) behaviour. Forest fires have also been proved to exhibit time-clustering phenomena, with timescales of the order of few days [3]. In this paper, we study forest fires in the perspective of dynamical systems and fractional calculus (FC). Public domain forest fires catalogues, containing data of events occurred in Portugal, in the period 1980 up to 2011, are considered. The data is analysed in an annual basis, modelling the occurrences as sequences of Dirac impulses. The frequency spectra of such signals are determined using Fourier transforms, and approximated through PL trendlines. The PL parameters are then used to unveil the fractional-order dynamics characteristics of the data. To complement the analysis, correlation indices are used to compare and find possible relationships among the data. It is shown that the used approach can be useful to expose hidden patterns not captured by traditional tools.

Entropy analysis of DNA code dynamics in human chromosomes

Deoxyribonucleic acid, or DNA, is the most fundamental aspect of life but present day scientific knowledge has merely scratched the surface of the problem posed by its decoding. While experimental methods provide insightful clues, the adoption of analysis tools supported by the formalism of mathematics will lead to a systematic and solid build-up of knowledge. This paper studies human DNA from the perspective of system dynamics. By associating entropy and the Fourier transform, several global properties of the code are revealed. The fractional order characteristics emerge as a natural consequence of the information content. These properties constitute a small piece of scientific knowledge that will support further efforts towards the final aim of establishing a comprehensive theory of the phenomena involved in life.

Analysis of temperature time-series: embedding dynamics into the MDS method

Global warming and the associated climate changes are being the subject of intensive research due to their major impact on social, economic and health aspects of the human life. Surface temperature time-series characterise Earth as a slow dynamics spatiotemporal system, evidencing long memory behaviour, typical of fractional order systems. Such phenomena are difficult to model and analyse, demanding for alternative approaches. This paper studies the complex correlations between global temperature time-series using the Multidimensional scaling (MDS) approach. MDS provides a graphical representation of the pattern of climatic similarities between regions around the globe. The similarities are quantified through two mathematical indices that correlate the monthly average temperatures observed in meteorological stations, over a given period of time. Furthermore, time dynamics is analysed by performing the MDS analysis over slices sampling the time series. MDS generates maps describing the stations’ locus in the perspective that, if they are perceived to be similar to each other, then they are placed on the map forming clusters. We show that MDS provides an intuitive and useful visual representation of the complex relationships that are present among temperature time-series, which are not perceived on traditional geographic maps. Moreover, MDS avoids sensitivity to the irregular distribution density of the meteorological stations.

Dynamical modelling of a genetic algorithm

This work addresses the signal propagation and the fractional-order dynamics during the evolution of a genetic algorithm (GA). In order to investigate the phenomena involved in the GA population evolution, the mutation is exposed to excitation perturbations during some generations and the corresponding fitness variations are evaluated. Three distinct fitness functions are used to study their influence in the GA dynamics. The input and output signals are studied revealing a fractional-order dynamic evolution, characteristic of a long-term system memory.

New findings on the dynamics of HIV and TB coinfection models

In this paper we study a model for HIV and TB coinfection. We consider the integer order and the fractional order versions of the model. Let α∈[0.78,1.0] be the order of the fractional derivative, then the integer order model is obtained for α=1.0. The model includes vertical transmission for HIV and treatment for both diseases. We compute the reproduction number of the integer order model and HIV and TB submodels, and the stability of the disease free equilibrium. We sketch the bifurcation diagrams of the integer order model, for variation of the average number of sexual partners per person and per unit time, and the tuberculosis transmission rate. We analyze numerical results of the fractional order model for different values of α, including α=1. The results show distinct types of transients, for variation of α. Moreover, we speculate, from observation of the numerical results, that the order of the fractional derivative may behave as a bifurcation parameter for the model. We conclude that the dynamics of the integer and the fractional order versions of the model are very rich and that together these versions may provide a better understanding of the dynamics of HIV and TB coinfection.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

Analysis of Forest Fires by means of Pseudo Phase Plane and Multidimensional Scaling Methods

Forest fires dynamics is often characterized by the absence of a characteristic length-scale, long range correlations in space and time, and long memory, which are features also associated with fractional order systems. In this paper a public domain forest fires catalogue, containing information of events for Portugal, covering the period from 1980 up to 2012, is tackled. The events are modelled as time series of Dirac impulses with amplitude proportional to the burnt area. The time series are viewed as the system output and are interpreted as a manifestation of the system dynamics. In the first phase we use the pseudo phase plane (PPP) technique to describe forest fires dynamics. In the second phase we use multidimensional scaling (MDS) visualization tools. The PPP allows the representation of forest fires dynamics in two-dimensional space, by taking time series representative of the phenomena. The MDS approach generates maps where objects that are perceived to be similar to each other are placed on the map forming clusters. The results are analysed in order to extract relationships among the data and to better understand forest fires behaviour.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

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.

New Concepts Towards the Synthesis of Digital Circuits Through Genetic Algorithms

This paper analyses the performance of a Genetic Algorithm using two new concepts, namely a static fitness function including a discontinuity measure and a fractional-order dynamic fitness function, for the synthesis of combinational logic circuits. In both cases, experiments reveal superior results in terms of speed and convergence to achieve a solution.

Analysis of Two Arms Working in Cooperation

This paper analyzes the performance of two cooperative robot manipulators. In order to capture the working performancewe formulated several performance indices that measure the manipulability, the effort reduction and the equilibrium between the two robots. In this perspective the proposed indices we determined the optimal values for the system parameters. Furthermore, it is studied the implementation of fractional-order algorithms in the position/force control of two cooperative robotic manipulators holding an object.

Aplicação das FPAA na realização de sistemas de ordem fraccionária

Esta tese de dissertação tem como principal objetivo a implementação de controladores fracionários utilizando diapositivos analógicos FPAA (Field Programable Analog Array). Embora estes dispositivos já não sejam um tecnologia recente, não tiveram grande aceitação comercial, daí não ter sido grande a sua evolução nesta última década. Mas para a elaboração de alguns circuitos analógicos, nomeadamente filtros, amplificadores e mesmo controladores PID (Proporcional-Integrativo-Derivativo) analógicos torna-se numa ferramenta que pode facilitar o projeto e implementação. Para a realização deste estudo, utilizou-se a placa de desenvolvimento da Anadigm AN231K04-DVLP3 juntamente com o software disponibilizado pela mesma empresa, o AnadigmDesigner2. Para a simulação e observação dos resultados foi utilizada a DAQ (Data Acquisition) Hilink da Zelton juntamente com o software Matlab. De forma a testar a implementação dos controladores fracionários nas FPAA foram realizados alguns circuitos no software e enviados para a FPAA comparando os resultados obtidos na simulação com os visualizados no osciloscópio. Por último foi projetado um controlador PIlDm recorrendo aos métodos de aproximação inteira descritos neste documento implementados na FPAA recorrendo ao uso de filtros de primeira e segunda ordem.

Fractional generalization of memristor and higher order elements

Fractional calculus generalizes integer order derivatives and integrals. Memristor systems generalize the notion of electrical elements. Both concepts were shown to model important classes of phenomena. This paper goes a step further by embedding both tools in a generalization considering complex-order objects. Two complex operators leading to real-valued results are proposed. The proposed class of models generate a broad universe of elements. Several combinations of values are tested and the corresponding dynamical behavior is analyzed.

Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators

We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?

Robotic manipulators with vibrations: short time fourier transform of fractional spectra

This paper analyzes the signals captured during impacts and vibrations of a mechanical manipulator. In order to acquire and study the signals an experimental setup is implemented. The signals are treated through signal processing tools such as the fast Fourier transform and the short time Fourier transform. The results show that the Fourier spectrum of several signals presents a non integer behavior. The experimental study provides valuable results that can assist in the design of a control system to deal with the unwanted effects of vibrations.