Symbolic fractional dynamics

Fractional dynamics reveals long range memory properties of systems described by means of signals represented by real numbers. Alternatively, dynamical systems and signals can adopt a representation where states are quantified using a set of symbols. Such signals occur both in nature and in man made processes and have the potential of a aftermath as relevant as the classical counterpart. This paper explores the association of Fractional calculus and symbolic dynamics. The results are visualized by means of the multidimensional technique and reveal the association between the fractal dimension and one definition of fractional derivative.

Complex dynamics of financial indices

This paper presents a novel method for the analysis of nonlinear financial and economic systems. The modeling approach integrates the classical concepts of state space representation and time series regression. The analytical and numerical scheme leads to a parameter space representation that constitutes a valid alternative to represent the dynamical behavior. The results reveal that business cycles can be clearly revealed, while the noise effects common in financial indices can elegantly be filtered out of the results.

Fractional dynamics in financial indexes

The goal of this study is the analysis of the dynamical properties of financial data series from 32 worldwide stock market indices during the period 2000–2009 at a daily time horizon. Stock market indices are examples of complex interacting systems for which a huge amount of data exists. The methods and algorithms that have been explored for the description of physical phenomena become an effective background in the analysis of economical data. In this perspective are applied the classical concepts of signal analysis, Fourier transform and methods of fractional calculus. The results reveal classification patterns typical of fractional dynamical systems.

Dynamics of the Dow Jones and the NASDAQ stock indexes

The goal of this study is the analysis of the dynamical properties of financial data series from worldwide stock market indices. We analyze the Dow Jones Industrial Average ( ∧ DJI) and the NASDAQ Composite ( ∧ IXIC) indexes at a daily time horizon. The methods and algorithms that have been explored for description of physical phenomena become an effective background, and even inspiration, for very productive methods used in the analysis of economical data. We start by applying the classical concepts of signal analysis, Fourier transform, and methods of fractional calculus. In a second phase we adopt a pseudo phase plane approach.

Simulation and dynamics of freeway traffic

This paper presents the new package entitled Simulator of Intelligent Transportation Systems (SITS) and a computational oriented analysis of traffic dynamics. The SITS adopts a microscopic simulation approach to reproduce real traffic conditions considering different types of vehicles, drivers and roads. A set of experiments with the SITS reveal the dynamic phenomena exhibited by this kind of system. For this purpose a modelling formalism is developed that embeds the statistics and the Laplace transform. The results make possible the adoption of classical system theory tools and point out that it is possible to study traffic systems taking advantage of the knowledge gathered with automatic control algorithms. A complementary perspective for the analysis of the traffic flow is also quantified through the entropy measure.

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.

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.

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.

Dynamical analysis of compositions

This paper analyzes musical opus from the point of view of two mathematical tools, namely the entropy and the multidimensional scaling (MDS). The Fourier analysis reveals a fractional dynamics, but the time rhythm variations are diluted along the spectrum. The combination of time-window entropy and MDS copes with the time characteristics and is well suited to treat a large volume of data. The experiments focus on a large number of compositions classified along three sets of musical styles, namely “Classical”, “Jazz”, and “Pop & Rock” compositions. Without lack of generality, the present study describes the application of the tools and the sets of musical compositions in a methodology leading to clear conclusions, but extensions to other possibilities are straightforward. The results reveal significant differences in the musical styles, demonstrating the feasibility of the proposed strategy and motivating further developments toward a dynamical analysis of musical compositions.

Fractional dynamics in DNA

This paper addresses the DNA code analysis in the perspective of dynamics and fractional calculus. Several mathematical tools are selected to establish a quantitative method without distorting the alphabet represented by the sequence of DNA bases. The association of Gray code, Fourier transform and fractional calculus leads to a categorical representation of species and chromosomes.

Análise dinâmica de mercados financeiros

Na sociedade actual, é cada vez mais difícil desassociar o ambiente financeiro do ambiente social, tendo o primeiro influência directa ou indirecta em praticamente todos os aspectos da sociedade. A esta influência está associada a vasta quantidade de informação e serviços financeiros que possibilitam uma melhor compreensão do ambiente socioeconómico actual, permitindo também o estudo das evoluções e das dinâmicas dos mercados financeiros. Este trabalho refere-se ao estudo e comparação de algumas ferramentas disponíveis para a análise dinâmica e tentativa de previsão de alguns índices de bolsa escolhidos. Tais métodos a estudar são modelos clássicos como o Autoregressivo, Média Móvel e o Modelo Misto apresentado por Box e Jenkins. São também propostos dois métodos que tentam distanciar-se dos métodos tradicionais por apenas considerarem para a sua previsão os momentos semelhantes ao momento actual que se tenta prever, ao invés de considerar todo o espectro dos dados disponíveis, tal como os métodos clássicos referidos anteriormente.

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.

Multidimensional scaling analysis of the dynamics of a country economy

This paper analyzes the Portuguese short-run business cycles over the last 150 years and presents the multidimensional scaling (MDS) for visualizing the results. The analytical and numerical assessment of this long-run perspective reveals periods with close connections between the macroeconomic variables related to government accounts equilibrium, balance of payments equilibrium, and economic growth. The MDS method is adopted for a quantitative statistical analysis. In this way, similarity clusters of several historical periods emerge in the MDS maps, namely, in identifying similarities and dissimilarities that identify periods of prosperity and crises, growth, and stagnation. Such features are major aspects of collective national achievement, to which can be associated the impact of international problems such as the World Wars, the Great Depression, or the current global financial crisis, as well as national events in the context of broad political blueprints for the Portuguese society in the rising globalization process.

Fractional dynamics of genetic algorithms using hexagonal space tessellation

The paper formulates a genetic algorithm that evolves two types of objects in a plane. The fitness function promotes a relationship between the objects that is optimal when some kind of interface between them occurs. Furthermore, the algorithm adopts an hexagonal tessellation of the two-dimensional space for promoting an efficient method of the neighbour modelling. The genetic algorithm produces special patterns with resemblances to those revealed in percolation phenomena or in the symbiosis found in lichens. Besides the analysis of the spacial layout, a modelling of the time evolution is performed by adopting a distance measure and the modelling in the Fourier domain in the perspective of fractional calculus. The results reveal a consistent, and easy to interpret, set of model parameters for distinct operating conditions.

Root-locus practical sketching rules for fractional-order system

For integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.