Fractional order modelling of dynamic backlash

This paper studies the dynamical properties of systems with backlash and impact phenomena. This type of non-linearity can be tackled in the perspective of the fractional calculus theory. Fractional and integer order models are compared and their influence upon the emerging dynamics is analysed. It is demonstrated that fractional models can memorize dynamical effects due to multiple micro-collisions.

Seroprevalência da Brucelose nos últimos cinco anos na Unidade Local de Saúde do Alto-Minho

A brucelose é uma zoonose com elevada importância, causada por bactérias gram-negativas que são altamente patogénicas para uma grande variedade de animais e humanos. Existem zonas endémicas onde esta se prolifera com mais facilidade. Neste estudo os dados são relativos ao distrito de Viana do Castelo, os dados são recolhidas da base de dados da Unidade Local de Saúde do Alto-Minho, uma zona não considerada endémica. Os animais infetados são a principal fonte de contaminação e dispersão da brucelose, é necessário uma reduzida carga bacteriana para ocorrer a infeção. Trata-se de uma doença que está longe de ser erradicada, impondo-se tomar medidas preventivas em relação à contaminação. Os testes usados na sua deteção podem ser alterados e melhorados de acordo com o estádio da doença. Na ULSAM são usados o teste de Wright e eventualmente a pesquisa microbiológica da bactéria Brucella. É pertinente saber o número de testes positivos que ocorrem por ano, se existe alguma sazonalidade relacionada com a doença, assim como, relacionar os parâmetros bioquímicos com um teste de Wright positivo. Os dados foram recolhidos entre o ano 2009-2013 com um número total de testes de 1035, dos quais o número total de positivos para o teste são 102, mas apenas trinta são positivos com significância. Os dados foram recolhidos através do programa Clinidata utilizado como base de armazenamento de dados da ULSAM e foram tratados estatisticamente com o programa SPSS juntamente com o Excel. Este estudo permitiu concluir que o número de casos em 2009 e 2010 era superior aos restantes anos, o que descreve uma tendência para diminuição do número de casos de brucelose atualmente no distrito de Viana do Castelo. Em relação a sazonalidade, os meses que apresentam uma percentagem superior a 50% em relação seroprevalência são os meses de Junho, Novembro e Dezembro. Os resultados revelam como declarado pela Organização Mundial de Saúde que o Distrito de Viana do Castelo não é uma zona endémica. Através da análise estatística foi possível concluir que um dos parâmetros bioquímicos, neste caso o número de leucócitos, poderá estar diretamente relacionado com um teste de Wright positivo, uma vez que, 37% da amostra de testes positivos revelam leucopenia.

Numerical analysis of the initial conditions in fractional systems

Fractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of adequate values for the initial conditions in numerical simulations.

Fractional Order Generalized Information

This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Condition-Based Diagnosis of Mechatronic Systems Using a Fractional Calculus Approach

While fractional calculus (FC) is as old as integer calculus, its application has been mainly restricted to mathematics. However, many real systems are better described using FC equations than with integer models. FC is a suitable tool for describing systems characterised by their fractal nature, long-term memory and chaotic behaviour. It is a promising methodology for failure analysis and modelling, since the behaviour of a failing system depends on factors that increase the model’s complexity. This paper explores the proficiency of FC in modelling complex behaviour by tuning only a few parameters. This work proposes a novel two-step strategy for diagnosis, first modelling common failure conditions and, second, by comparing these models with real machine signals and using the difference to feed a computational classifier. Our proposal is validated using an electrical motor coupled with a mechanical gear reducer.

Fractional order describing functions

This paper addresses limit cycles and signal propagation in dynamical systems with backlash. The study follows the describing function (DF) method for approximate analysis of nonlinearities and generalizes it in the perspective of the fractional calculus. The concept of fractional order describing function (FDF) is illustrated and the results for several numerical experiments are analysed. FDF leads to a novel viewpoint for limit cycle signal propagation as time-space waves within system structure.

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.

Pseudo Phase Plane and Fractional Calculus Modelling of Western Global Economic Downturn

This paper applies Pseudo Phase Plane (PPP) and Fractional Calculus (FC) mathematical tools for modeling world economies. A challenging global rivalry among the largest international economies began in the early 1970s, when the post-war prosperity declined. It went on, up to now. If some worrying threatens may exist actually in terms of possible ambitious military aggression, invasion, or hegemony, countries’ PPP relative positions can tell something on the current global peaceful equilibrium. A global political downturn of the USA on global hegemony in favor of Asian partners is possible, but can still be not accomplished in the next decades. If the 1973 oil chock has represented the beginning of a long-run recession, the PPP analysis of the last four decades (1972–2012) does not conclude for other partners’ global dominance (Russian, Brazil, Japan, and Germany) in reaching high degrees of similarity with the most developed world countries. The synergies of the proposed mathematical tools lead to a better understanding of the dynamics underlying world economies and point towards the estimation of future states based on the memory of each time series.

Fractional Calculus: Quo Vadimus? (Where Are We Going?)

Discussions under this title were held during a special session in frames of the International Conference “Fractional Differentiation and Applications” (ICFDA ’14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.

Integer/fractional decomposition of the impulse response of fractional linear systems

The decomposition of a fractional linear system is discussed in this paper. It is shown that it can be decomposed into an integer order part, corresponding to possible existing poles, and a fractional part. The first and second parts are responsible for the short and long memory behaviors of the system, respectively, known as characteristic of fractional systems.

A Fractional Perspective to the Bond Graph Modelling of World Economies

Inspired in dynamic systems theory and Brewer’s contributions to apply it to economics, this paper establishes a bond graph model. Two main variables, a set of inter-connectivities based on nodes and links (bonds) and a fractional order dynamical perspective, prove to be a good macro-economic representation of countries’ potential performance in nowadays globalization. The estimations based on time series for 50 countries throughout the last 50 decades confirm the accuracy of the model and the importance of scale for economic performance.

Numerical calculation of the left and right fractional derivatives

The calculation of fractional derivatives is an important topic in scientific research. While formal definitions are clear from the mathematical point of view, they pose limitations in applied sciences that have not been yet tackled. This paper addresses the problem of obtaining left and right side derivatives when adopting numerical approximations. The results reveal the relationship between the resulting distinct values for different fractional orders and types of signals.