A coherent approach to non-integer order derivatives

Signal Processing, vol. 86, nº 10

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Starting from zero-, first-, and second-order integrated laws for chemical kinetics, some cases are shown which produce fractional orders. Taking the Michaelis-Menten mechanism as a first example, it is shown that substrate order can go from 1 to zero, depending on relative concentration of enzyme and substrate. Using other examples which show fractional orders higher than one and even negative (inhibition), it is shown that the presence of an equilibrium before or parallel to the rate determining step can be the reason for fractional orders, which is an indication of a more complex mechanism.

Fractional dynamics and MDS visualization of earthquake phenomena

This paper analyses earthquake data in the perspective of dynamical systems and fractional calculus (FC). This new standpoint uses Multidimensional Scaling (MDS) as a powerful clustering and visualization tool. FC extends the concepts of integrals and derivatives to non-integer and complex orders. MDS is a technique that produces spatial or geometric representations of complex objects, such that those objects that are perceived to be similar in some sense are placed on the MDS maps forming clusters. In this study, over three million seismic occurrences, covering the period from January 1, 1904 up to March 14, 2012 are analysed. The events are characterized by their magnitude and spatiotemporal distributions and are divided into fifty groups, according to the Flinn–Engdahl (F–E) seismic regions of Earth. Several correlation indices are proposed to quantify the similarities among regions. MDS maps are proven as an intuitive and useful visual representation of the complex relationships that are present among seismic events, which may not be perceived on traditional geographic maps. Therefore, MDS constitutes a valid alternative to classic visualization tools for understanding the global behaviour of earthquakes.

Analysis and implementation of a new constant acceleration subcycling algorithm

An algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps. With integer time step ratios, the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle. The algorithm is shown to have good accuracy, and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms. The implementation of a generalised form of the algorithm with non-integer time step ratios is presented. (C) 1997 by John Wiley & Sons, Ltd.

Fractional central pattern generators for bipedal locomotion

Locomotion has been a major research issue in the last few years. Many models for the locomotion rhythms of quadrupeds, hexapods, bipeds and other animals have been proposed. This study has also been extended to the control of rhythmic movements of adaptive legged robots. In this paper, we consider a fractional version of a central pattern generator (CPG) model for locomotion in bipeds. A fractional derivative D α f(x), with α non-integer, is a generalization of the concept of an integer derivative, where α=1. The integer CPG model has been proposed by Golubitsky, Stewart, Buono and Collins, and studied later by Pinto and Golubitsky. It is a network of four coupled identical oscillators which has dihedral symmetry. We study parameter regions where periodic solutions, identified with legs’ rhythms in bipeds, occur, for 0<α≤1. We find that the amplitude and the period of the periodic solutions, identified with biped rhythms, increase as α varies from near 0 to values close to unity.

On the fractional order control of heat systems

The differentiation of non-integer order has its origin in the seventeenth century, but only in the last two decades appeared the first applications in the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated namely the fractional PID and the Smith predictor. Extensive simulations are presented assessing the performance of the proposed fractional-order algorithms.

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.

Riesz potential operators and inverses via fractional centred derivatives

International Journal of Mathematics and Mathematical Sciences, Vol.2006

Fractional control of heat diffusion systems

The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared. Simulations are presented assessing the performance of the proposed fractional-order algorithms.

Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Rhapsody in Fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Bipedal Locomotion: A Fractional CPG for Generating Patterns

Proceedings of the 10th Conference on Dynamical Systems Theory and Applications

Une analyse de la sanction économique en droit international

Le principal reproche qui est adressé au droit international est la faiblesse de ses mécanismes de sanctions. Pour cette raison, plusieurs penseurs juridiques ont conclu que le droit international n'existait pas. Le présent mémoire vise donc à étudier la rhétorique derrière cette affirmation et à examiner sa validité. Pour ce faire, nous analysons dans un premier temps la relation entre la sanction et le droit à travers le cadre positiviste du XIXe siècle. Nous étudions tour à tour les soi-disant critères constitutifs de la sanction. Nous les comparons avec d'autres ordres non juridiques pour ensuite rejeter le postulat positiviste qui fait de la contrainte et de la centralisation des pouvoirs les éléments essentiels de la sanction. Nous étudions ensuite le cadre d'adoption des sanctions économiques du Conseil de sécurité des Nations Unies. Cette étape nous permet d'expliquer les principales faiblesses et limitations du droit international. Parmi celles-ci, le manque de coopération internationale, le droit de veto et le principe de la souveraineté nationale sont les éléments qui freinent l'adoption des sanctions coopération internationale. Nous examinons ensuite les objectifs derrière l'imposition des sanctions ainsi que leur efficacité. Finalement, nous étudions les embargos en général et les embargos sur les armes. Cette étude nous permet d'une part de démontrer les effets des sanctions économiques sur la population civile et sur les États tiers. D'autre part, elle nous permet de mieux comprendre les problèmes relatifs à l'administration d'une sanction, ainsi qu'aux mesures de contournement des interdictions.