Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

A review of sample and hold systems and design of a new fractional algorithm

UIDB/00066/2020Digital systems require sample and hold (S&H) systems to perform the conversion from analog to digital and vice versa. Besides the standard zero and first order holds, we find in the literature other versions, namely the fractional and exponential order holds, involving parameters that can be tuned to produce a superior performance. This paper reviews the fundamental concepts associated with the S&H and proposes a new fractional version. The systems are modeled both in the time and Laplace domains. The new S&H stemming from fractional calculus generalizes these devices. The different S&H systems are compared in the frequency domain and their relationships visualized by means of hierarchical clustering and multidimensional scaling representations. The novel strategy allows a better understanding of the possibilities and limitations of S&H systems.publishersversionpublishe

Revisiting the 1D and 2D laplace transforms

Foundation for Science and Technology of Portugal, under the projects UIDB/00066/2020.The paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined.publishersversionpublishe

Existence of Bounded Solutions to a Modified Version of the Bagley–Torvik Equation

This manuscript reanalyses the Bagley–Torvik equation (BTE). The Riemann–Liouville fractional differential equation (FDE), formulated by R. L. Bagley and P. J. Torvik in 1984, models the vertical motion of a thin plate immersed in a Newtonian fluid, which is held by a spring. From this model, we can derive an FDE for the particular case of lacking the spring. Here, we find conditions for the source term ensuring that the solutions to the equation of the motion are bounded, which has a clear physical meaning.S

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.info:eu-repo/semantics/publishedVersio

On the Numerical Computation of the Mittag-Leffler Function

The Mittag-Leffler function (MLF) plays an important role in many applications of fractional calculus, establishing a connection between exponential and power law behaviors that characterize integer and fractional order phenomena, respectively. Nevertheless, the numerical computation of the MLF poses problems both of accuracy and convergence. In this paper, we study the calculation of the 2-parameter MLF by using polynomial computation and integral formulas. For the particular cases having Laplace transform (LT) the method relies on the inversion of the LT using the fast Fourier transform. Experiments with two other available methods compare also the computational time and accuracy. The 3-parameter MLF and its calculation are also considered.publishe

A Clustering Perspective of the Collatz Conjecture

This manuscript focuses on one of the most famous open problems in mathematics, namely the Collatz conjecture. The first part of the paper is devoted to describe the problem, providing a historical introduction to it, as well as giving some intuitive arguments of why is it hard from the mathematical point of view. The second part is dedicated to the visualization of behaviors of the Collatz iteration function and the analysis of the resultsThe work of D. Cao Labora was partially supported by grant number MTM2016-75140-P (AEI/FEDER, UE)S

Análise de sinais em manipuladores com membros flexíveis

Neste trabalho apresenta-se a análise de sinais da dinâmica de uma estrutura robótica com membros flexíveis. A estrutura mecânica deste manipulador é composta de membros constituídos de segmentos rígidos interligados por juntas rotacionais. Elabora-se a dinâmica inversa utilizando-se uma formulação numérica recursiva de Newton-Euler, semelhante à utilizada para sistemas rígidos. Pela manipulação adequada de vectores de entrada, na rotina de estruturação da dinâmica inversa, obtém-se a matriz de massa e os vectores relacionados com a acção da gravidade, de Coriolis e centrípeta. Assim. obtém-se o sistema dinâmico directo da estrutura discretizada com um número adequado de graus de liberdade ngl. Impõe-se no eixo do segmento activo de um membro um binário perturbador de excitação senoidal. Pela dinâmica directa faz-se a simulação, com discretização temporal, desse sistema utilizando métodos de integração númerica. Obtém-se a resposta temporal dos vectores deslocamento, velocidade e aceleração de todas as juntas. Através do MatLab os dados são tratados para analisar as funções de resposta no tempo e em frequência. São apresentados alguns resultados obtidos com algumas simulações realizadas.N/

A Fourier Perspective of Freeway Traffic

This paper presents the Simulator of Intelligent Transportation Systems (SITS) and a dynamical analysis of several traffic phenomena. The SITS is based on a microscopic simulation approach to reproduce real traffic and considers different types of vehicles, drivers and roads. In order to process the resulting information, it is also introduced a new modelling scheme based on the embedding of statistics and Fourier transform. The results point out that it is possible to study traffic systems, taking advantage on the knowledge gathered with systems theory.N/