Entropy analysis of integer and fractional dynamical systems

This paper investigates the adoption of entropy for analyzing the dynamics of a multiple independent particles system. Several entropy definitions and types of particle dynamics with integer and fractional behavior are studied. The results reveal the adequacy of the entropy concept in the analysis of complex dynamical systems.

Controling delayed transitions with applications to prevent single species extinctions

A new method is proposed to control delayed transitions towards extinction in single population theoretical models with discrete time undergoing saddle-node bifurcations. The control method takes advantage of the delaying properties of the saddle remnant arising after the bifurcation, and allows to sustain populations indefinitely. Our method, which is shown to work for deterministic and stochastic systems, could generally be applied to avoid transitions tied to one-dimensional maps after saddle-node bifurcations.

Generalized models from Beta(p,2) densities with strong allee effect: dynamical approach

A dynamical approach to study the behaviour of generalized populational growth models from Bets(p, 2) densities, with strong Allee effect, is presented. The dynamical analysis of the respective unimodal maps is performed using symbolic dynamics techniques. The complexity of the correspondent discrete dynamical systems is measured in terms of topological entropy. Different populational dynamics regimes are obtained when the intrinsic growth rates are modified: extinction, bistability, chaotic semistability and essential extinction.

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.

Dynamical modelling of a genetic algorithm

This work addresses the signal propagation and the fractional-order dynamics during the evolution of a genetic algorithm (GA). In order to investigate the phenomena involved in the GA population evolution, the mutation is exposed to excitation perturbations during some generations and the corresponding fitness variations are evaluated. Three distinct fitness functions are used to study their influence in the GA dynamics. The input and output signals are studied revealing a fractional-order dynamic evolution, characteristic of a long-term system memory.

Exciting dynamical behavior in a network of two coupled rings of Chen oscillators

We study exotic patterns appearing in a network of coupled Chen oscillators. Namely, we consider a network of two rings coupled through a “buffer” cell, with Z3×Z5 symmetry group. Numerical simulations of the network reveal steady states, rotating waves in one ring and quasiperiodic behavior in the other, and chaotic states in the two rings, to name a few. The different patterns seem to arise through a sequence of Hopf bifurcations, period-doubling, and halving-period bifurcations. The network architecture seems to explain certain observed features, such as equilibria and the rotating waves, whereas the properties of the chaotic oscillator may explain others, such as the quasiperiodic and chaotic states. We use XPPAUT and MATLAB to compute numerically the relevant states.

Low temperature structural transitions in dipolar hard spheres: the influence on magnetic properties

We investigate the structural chain-to-ring transition at low temperature in a gas of dipolar hard spheres (DRS). Due to the weakening of entropic contribution, ring formation becomes noticeable when the effective dipole-dipole magnetic interaction increases, It results in the redistribution of particles from usually observed flexible chains into flexible rings. The concentration (rho) of DI-IS plays a crucial part in this transition: at a very low rho only chains and rings are observed, whereas even a slight increase of the volume fraction leads to the formation of branched or defect structures. As a result, the fraction of DHS aggregated in defect-free rings turns out to be a non-monotonic function of rho. The average ring size is found to be a slower increasing function of rho when compared Lo that of chains. Both theory and computer simulations confirm the dramatic influence of the ring formation on the rho-dependence of the initial magnetic susceptibility (chi) when the temperature decreases. The rings clue to their zero total dipole moment are irresponsive to a weak magnetic field and drive to the strong decrease of the initial magnetic susceptibility. (C) 2014 Elsevier B.V. All rights reserved.

Temperature-induced structural transitions in self-assembling magnetic nanocolloids

With the help of a unique combination of density functional theory and computer simulations, we discover two possible scenarios, depending on concentration, for the hierarchical self-assembly of magnetic nanoparticles on cooling. We show that typically considered low temperature clusters, i.e. defect-free chains and rings, merge into more complex branched structures through only three types of defects: four-way X junctions, three-way Y junctions and two-way Z junctions. Our accurate calculations reveal the predominance of weakly magnetically responsive rings cross-linked by X defects at the lowest temperatures. We thus provide a strategy to fine-tune magnetic and thermodynamic responses of magnetic nanocolloids to be used in medical and microfluidics applications.

Spectral invariants of periodic nonautonomous discrete dynamical systems

For an interval map, the poles of the Artin-Mazur zeta function provide topological invariants which are closely connected to topological entropy. It is known that for a time-periodic nonautonomous dynamical system F with period p, the p-th power [zeta(F) (z)](p) of its zeta function is meromorphic in the unit disk. Unlike in the autonomous case, where the zeta function zeta(f)(z) only has poles in the unit disk, in the p-periodic nonautonomous case [zeta(F)(z)](p) may have zeros. In this paper we introduce the concept of spectral invariants of p-periodic nonautonomous discrete dynamical systems and study the role played by the zeros of [zeta(F)(z)](p) in this context. As we will see, these zeros play an important role in the spectral classification of these systems.

Dynamical Stability and Predictability of Football Players: The Study of One Match

The game of football demands new computational approaches to measure individual and collective performance. Understanding the phenomena involved in the game may foster the identification of strengths and weaknesses, not only of each player, but also of the whole team. The development of assertive quantitative methodologies constitutes a key element in sports training. In football, the predictability and stability inherent in the motion of a given player may be seen as one of the most important concepts to fully characterise the variability of the whole team. This paper characterises the predictability and stability levels of players during an official football match. A Fractional Calculus (FC) approach to define a player’s trajectory. By applying FC, one can benefit from newly considered modeling perspectives, such as the fractional coefficient, to estimate a player’s predictability and stability. This paper also formulates the concept of attraction domain, related to the tactical region of each player, inspired by stability theory principles. To compare the variability inherent in the player’s process variables (e.g., distance covered) and to assess his predictability and stability, entropy measures are considered. Experimental results suggest that the most predictable player is the goalkeeper while, conversely, the most unpredictable players are the midfielders. We also conclude that, despite his predictability, the goalkeeper is the most unstable player, while lateral defenders are the most stable during the match.

Spectral and dynamical invariants in a complete clustered network

The main result of this work is a new criterion for the formation of good clusters in a graph. This criterion uses a new dynamical invariant, the performance of a clustering, that characterizes the quality of the formation of clusters. We prove that the growth of the dynamical invariant, the network topological entropy, has the effect of worsening the quality of a clustering, in a process of cluster formation by the successive removal of edges. Several examples of clustering on the same network are presented to compare the behavior of other parameters such as network topological entropy, conductance, coefficient of clustering and performance of a clustering with the number of edges in a process of clustering by successive removal.

Dynamical Analysis and Visualization of Tornadoes Time Series

In this paper we analyze the behavior of tornado time-series in the U.S. from the perspective of dynamical systems. A tornado is a violently rotating column of air extending from a cumulonimbus cloud down to the ground. Such phenomena reveal features that are well described by power law functions and unveil characteristics found in systems with long range memory effects. Tornado time series are viewed as the output of a complex system and are interpreted as a manifestation of its dynamics. Tornadoes are modeled as sequences of Dirac impulses with amplitude proportional to the events size. First, a collection of time series involving 64 years is analyzed in the frequency domain by means of the Fourier transform. The amplitude spectra are approximated by power law functions and their parameters are read as an underlying signature of the system dynamics. Second, it is adopted the concept of circular time and the collective behavior of tornadoes analyzed. Clustering techniques are then adopted to identify and visualize the emerging patterns.

Study of forbidden transitions in atomic systems

A Thesis submitted for the co-tutelle degree of Doctor in Physics at Universidade Nova de Lisboa and Université Pierre et Marie Curie

Dynamical Analysis of Non-Repetitive Trajectories in Robotic Manipulation

A new method for the study and optimization of manu«ipulator trajectories is developed. The novel feature resides on the modeling formulation. Standard system desciptions are based on a set of differential equations which, in general, require laborious computations and may be difficult to analyze. Moreover, the derived algorithms are suited to "deterministic" tasks, such as those appearing in a repetitivework, and are not well adapted to a "random" operation that occurs in intelligent systems interacting with a non-structured and changing environment. These facts motivate the development of alternative models based on distinct concepts. The proposed embedding of statistics and Fourier trasnform gives a new perspective towards the calculation and optimization of the robot trajectories in manipulating tasks.

Price-setting dynamical duopoly with incomplete information

We consider a price competition in a duopoly with substitutable goods, linear and symmetric demand. There is a firm (F 1) that chooses first the price p 1 of its good; the other firm (F 2) observes p 1 and then chooses the price p 2 of its good. The conclusions of this price-setting dynamical duopoly are substantially altered by the presence of either differentiated goods or asymmetric information about rival’s production costs. In this paper, we consider asymmetric information about rival’s production costs. We do ex-ante and ex-post analyses of firms’ profits and market prices. We compare the ex-ante firms’ expected profits with the ex-post firms’ profits.