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.

Big Bang bifurcations and allee effect in Blumberg’s dynamics

This paper concerns dynamics and bifurcations properties of a class of continuous-defined one-dimensional maps, in a three-dimensional parameter space: Blumberg's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon, associated with the stability of a fixed point. A central point of our investigation is the study of bifurcations structure for this class of functions. We verified that under some sufficient conditions, Blumberg's functions have a particular bifurcations structure: the big bang bifurcations of the so-called "box-within-a-box" type, but for different kinds of boxes. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct attractors. This work contributes to clarify the big bang bifurcation analysis for continuous maps. To support our results, we present fold and flip bifurcations curves and surfaces, and numerical simulations of several bifurcation diagrams.

Spatial dynamics of team sports exposed by Voronoi diagrams

Team sports represent complex systems: players interact continuously during a game, and exhibit intricate patterns of interaction, which can be identified and investigated at both individual and collective levels. We used Voronoi diagrams to identify and investigate the spatial dynamics of players' behavior in Futsal. Using this tool, we examined 19 plays of a sub-phase of a Futsal game played in a reduced area (20 m(2)) from which we extracted the trajectories of all players. Results obtained from a comparative analysis of player's Voronoi area (dominant region) and nearest teammate distance revealed different patterns of interaction between attackers and defenders, both at the level of individual players and teams. We found that, compared to defenders, larger dominant regions were associated with attackers. Furthermore, these regions were more variable in size among players from the same team but, at the player level, the attackers' dominant regions were more regular than those associated with each of the defenders. These findings support a formal description of the dynamic spatial interaction of the players, at least during the particular sub-phase of Futsal investigated. The adopted approach may be extended to other team behaviors where the actions taken at any instant in time by each of the involved agents are associated with the space they occupy at that particular time.

Fractional describing function of systems with nonlinear friction

This paper studies the describing function (DF) of systems consisting in a mass subjected to nonlinear friction. The friction force is composed in three components namely, the viscous, the Coulomb and the static forces. The system dynamics is analyzed in the DF perspective revealing a fractional-order behaviour. The reliability of the DF method is evaluated through the signal harmonic content and the limit cycle prediction.

Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.

Complex Dynamics of defective interfering baculoviruses during serial passage in insect cells

Defective interfering (DI) viruses are thought to cause oscillations in virus levels, known as the ‘Von Magnus effect’. Interference by DI viruses has been proposed to underlie these dynamics, although experimental tests of this idea have not been forthcoming. For the baculoviruses, insect viruses commonly used for the expression of heterologous proteins in insect cells, the molecular mechanisms underlying DI generation have been investigated. However, the dynamics of baculovirus populations harboring DIs have not been studied in detail. In order to address this issue, we used quantitative real-time PCR to determine the levels of helper and DI viruses during 50 serial passages of Autographa californica multiple nucleopolyhedrovirus (AcMNPV) in Sf21 cells. Unexpectedly, the helper and DI viruses changed levels largely in phase, and oscillations were highly irregular, suggesting the presence of chaos. We therefore developed a simple mathematical model of baculovirus-DI dynamics. This theoretical model reproduced patterns qualitatively similar to the experimental data. Although we cannot exclude that experimental variation (noise) plays an important role in generating the observed patterns, the presence of chaos in the model dynamics was confirmed with the computation of the maximal Lyapunov exponent, and a Ruelle-Takens-Newhouse route to chaos was identified at decreasing production of DI viruses, using mutation as a control parameter. Our results contribute to a better understanding of the dynamics of DI baculoviruses, and suggest that changes in virus levels over passages may exhibit chaos.

Complex dynamics in the trajectory control of redundant manipulators

Redundant manipulators allow the trajectory optimization, the obstacle avoidance, and the resolution of singularities. For this type of manipulators, the kinematic control algorithms adopt generalized inverse matrices that may lead to unpredictable responses. Motivated by these problems this paper studies the complexity revealed by the trajectory planning scheme when controlling redundant manipulators. The results reveal fundamental properties of the chaotic phenomena and give a deeper insight towards the development of superior trajectory control algorithms.

The contribution of multitemporal information from multispectral satellite images for automatic land cover classification at the national scale

Thesis submitted to the Instituto Superior de Estatística e Gestão de Informação da Universidade Nova de Lisboa in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Information Management – Geographic Information Systems

Fractional describing function of systems with Coulomb friction

This paper studies the describing function (DF) of systems constituted by a mass subjected to nonlinear friction. The friction force is decomposed into two components, namely, the viscous and the Coulomb friction. The system dynamics is analyzed in the DF perspective revealing a fractional-order behavior. The reliability of the DF method is evaluated through the signal harmonic contents.

Development of fractional order capacitors based on electrolyte processes

In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.

Dynamics of transmission of Trypanosoma cruzi in a rural area of Argentina: III. Persistence of T. cruzi parasitemia among canine reservoirs in a two-year follow-up

A new cross-sectional survey of household- associated mongrel dogs as well as follow-up of previously parasitemic individuals was carried out in 1984 toy means of xenodiagnosis and serologic techniques to get a deeper insight into the relationship of T. cruzi parasitemia and age among canine hosts in a rural area of Argentina. Persistence of detectable parasitemia was age-independent, or at most, loosely related to age, confirming the pattern observed in 1982. Similarly no significant age-decreasing effect was recorded among seropositive dogs in: a) the probability of detecting parasites in a 2-year follow-up; b) their intensity of infectiousness (=infective force) for T. infestans 3rd-4th instar nymphs, as measured by the percentage of infected bugs observed in each dog xenodiagnosis. Moreover, not only was the infective force of seropositive dogs for bugs approximately constant through lifetime, but it was significantly higher than the one recorded for children in the present survey, and for human people by other researchers. Therefore, and since T. infestans field populations show high feeding frequencies on dogs, the latter are expected to make the greatest contribution to the pool of infected vectors in the rural household of Argentina. This characteristic should be sufficient to involve canine reservoirs definitely as a risk factor for human people residing in the same house. The increased severity of parasitemia observed among dogs in this survey may be related to the acute undernutrition characteristic of canine populations of poor rural areas in our country, which is expected to affect the ability of the host to manage the infection.

Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.

Complex-order dynamics in hexapod locomotion

This paper studies the dynamics of foot–ground interaction in hexapod locomotion systems. For that objective the robot motion is characterized in terms of several locomotion variables and the ground is modelled through a non-linear spring-dashpot system, with parameters based on the studies of soil mechanics. Moreover, it is adopted an algorithm with foot-force feedback to control the robot locomotion. A set of model-based experiments reveals the influence of the locomotion velocity on the foot–ground transfer function, which presents complex-order dynamics.

Soil attributes dynamics evaluation after prescribed burning practice in northwestern Portugal forest

The Portuguese northern forests are often and severely affected by wildfires during the summer season. These occurrences affect significant and rudely all ecosystems, namely soil, fauna and flora. Preventive actions such as prescribed burnings and clear-cut logging are frequently used and have showed a significant reduction of the natural wildfires occurrences. In Portugal, and due to some technical and operational conditions, prescribed burnings in forests are the most common preventive action used to reduce the existing fuel hazard. The overall impacts of this preventive action on Portuguese ecosystems are complex and not fully understood. This work reports to the study of a prescribed burning impact in soil chemical properties, namely pH, humidity and organic matter, by monitoring the soil self-recovery capacity. The experiments were carried out in soil cover over a natural site of Andaluzitic schist, in Gramelas, Caminha, Portugal, who was able to maintain itself intact from prescribed burnings from four years. The composed soil samples were collected from five plots at three different layers (0-3cm, 3-6cm and 6-18cm) 1 day before prescribed fire and after the prescribed fire. The results have shown that the dynamic equilibrium in soil was affected significantly.

Management Driven Hybrid Multicast Framework for Content Aware Networks

The need for better adaptation of networks to transported flows has led to research on new approaches such as content aware networks and network aware applications. In parallel, recent developments of multimedia and content oriented services and applications such as IPTV, video streaming, video on demand, and Internet TV reinforced interest in multicast technologies. IP multicast has not been widely deployed due to interdomain and QoS support problems; therefore, alternative solutions have been investigated. This article proposes a management driven hybrid multicast solution that is multi-domain and media oriented, and combines overlay multicast, IP multicast, and P2P. The architecture is developed in a content aware network and network aware application environment, based on light network virtualization. The multicast trees can be seen as parallel virtual content aware networks, spanning a single or multiple IP domains, customized to the type of content to be transported while fulfilling the quality of service requirements of the service provider.