Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators

We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?

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.

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.

Complex order biped rhythms

Animal locomotion is a complex process, involving the central pattern generators (neural networks, located in the spinal cord, that produce rhythmic patterns), the brainstem command systems, the steering and posture control systems and the top layer structures that decide which motor primitive is activated at a given time. Pinto and Golubitsky studied an integer CPG model for legs rhythms in bipeds. It is a four-coupled identical oscillators' network with dihedral symmetry. This paper considers a new complex order central pattern generator (CPG) model for locomotion in bipeds. A complex derivative Dα±jβ, with α, β ∈ ℜ+, j = √-1, is a generalization of the concept of an integer derivative, where α = 1, β = 0. Parameter regions where periodic solutions, identified with legs' rhythms in bipeds, occur, are analyzed. Also observed is the variation of the amplitude and period of periodic solutions with the complex order derivative.

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.

Two-production-period in a duopoly with nonprofit and for-profit firms

We investigate endogenous roles in a competition between a nonprofit firm and a for-profit firm in a homogeneous goods market, by allowing two production periods. We find that the Cournot-type equilibrium and one Stackelberg-type equilibrium where the nonprofit firm becomes the follower exist; however, another tackelberg-type equilibrium where the nonprofit firm becomes the leader does not exist.

Forest fires occurrences and burned area in mainland Portugal: preliminary assessment to a fifteen years period

Portugal, as well as the Mediterranean basin, is favorable to the occurrence of forest fires. In this work a statistical analysis was carried out based on the official information, considering the forest fires occurrences and the corresponding burned area for each of the districts of the mainland Portugal, between 1996 and 2010. Concerning to the forest fires occurrence it was possible to identify three main regions in mainland Portugal, while the burned area can be characterized in two main regions. Associations between districts and years are different in the two approaches. The results obtained provide a synthetic analysis of the phenomenon of forest fires in continental Portugal, based on all the official information available to date.

Mathematical model for HIV dynamics in HIV-specific helper cells

In this paper we study a delay mathematical model for the dynamics of HIV in HIV-specific CD4 + T helper cells. We modify the model presented by Roy and Wodarz in 2012, where the HIV dynamics is studied, considering a single CD4 + T cell population. Non-specific helper cells are included as alternative target cell population, to account for macrophages and dendritic cells. In this paper, we include two types of delay: (1) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and; (2) virus production period for new virions to be produced within and released from the infected cells. We compute the reproduction number of the model, R0, and the local stability of the disease free equilibrium and of the endemic equilibrium. We find that for values of R0<1, the model approaches asymptotically the disease free equilibrium. For values of R0>1, the model approximates asymptotically the endemic equilibrium. We observe numerically the phenomenon of backward bifurcation for values of R0⪅1. This statement will be proved in future work. We also vary the values of the latent period and the production period of infected cells and free virus. We conclude that increasing these values translates in a decrease of the reproduction number. Thus, a good strategy to control the HIV virus should focus on drugs to prolong the latent period and/or slow down the virus production. These results suggest that the model is mathematically and epidemiologically well-posed.

Age-period-cohort effects in the incidence of hip fractures: political and economic events are coincident with changes in risk

Introduction: Healthcare improvements have allowed prevention but have also increased life expectancy, resulting in more people being at risk. Our aim was to analyse the separate effects of age, period and cohort on incidence rates by sex in Portugal, 2000–2008. Methods: From the National Hospital Discharge Register, we selected admissions (aged ≥49 years) with hip fractures (ICD9-CM, codes 820.x) caused by low/moderate trauma (falls from standing height or less), readmissions and bone cancer cases. We calculated person-years at risk using population data from Statistics Portugal. To identify period and cohort effects for all ages, we used an age–period–cohort model (1-year intervals) followed by generalised additive models with a negative binomial distribution of the observed incidence rates of hip fractures. Results: There were 77,083 hospital admissions (77.4 % women). Incidence rates increased exponentially with age for both sexes (age effect). Incidence rates fell after 2004 for women and were random for men (period effect). There was a general cohort effect similar in both sexes; risk of hip fracture altered from an increasing trend for those born before 1930 to a decreasing trend following that year. Risk alterations (not statistically significant) coincident with major political and economic change in the history of Portugal were observed around birth cohorts 1920 (stable–increasing), 1940 (decreasing–increasing) and 1950 (increasing–decreasing only among women). Conclusions: Hip fracture risk was higher for those born during major economically/politically unstable periods. Although bone quality reflects lifetime exposure, conditions at birth may determine future risk for hip fractures.

Power Law Behavior and Self-similarity in Modern Industrial Accidents

Advances in technology have produced more and more intricate industrial systems, such as nuclear power plants, chemical centers and petroleum platforms. Such complex plants exhibit multiple interactions among smaller units and human operators, rising potentially disastrous failure, which can propagate across subsystem boundaries. This paper analyzes industrial accident data-series in the perspective of statistical physics and dynamical systems. Global data is collected from the Emergency Events Database (EM-DAT) during the time period from year 1903 up to 2012. The statistical distributions of the number of fatalities caused by industrial accidents reveal Power Law (PL) behavior. We analyze the evolution of the PL parameters over time and observe a remarkable increment in the PL exponent during the last years. PL behavior allows prediction by extrapolation over a wide range of scales. In a complementary line of thought, we compare the data using appropriate indices and use different visualization techniques to correlate and to extract relationships among industrial accident events. This study contributes to better understand the complexity of modern industrial accidents and their ruling principles.

Multidimensional Scalling Analysis of the World Economy During the Period 1972-2012

The last 40 years of the world economy are analyzed by means of computer visualization methods. Multidimensional scaling and the hierarchical clustering tree techniques are used. The current Western downturn in favor of Asian partners may still be reversed in the coming decades.