Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincareacute return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing delta(1). However, for higher values of delta(1) the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter zeta) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

Measuring and Controlling the Chaotic Motion of Profits

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Toplological Entropy in the Syncronization of Piecewise Linear and Monotone Maps. Coupled Duffing Oscillators

In this paper is presented a relationship between the synchronization and the topological entropy. We obtain the values for the coupling parameter, in terms of the topological entropy, to achieve synchronization of two unidirectional and bidirectional coupled piecewise linear maps. In addition, we prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the unidirectional and bidirectional coupled identical chaotic Duffing equations is given. We discuss the complete synchronization of two identical double-well Duffing oscillators, from the point of view of symbolic dynamics. Working with Poincare cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

On the DNA of eleven mammals

This paper studies the DNA code of eleven mammals from the perspective of fractional dynamics. The application of Fourier transform and power law trendlines leads to a categorical representation of species and chromosomes. The DNA information reveals long range memory characteristics.

An Extension of Gompertzian Growth Dynamics Weibull and Frechet Models

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.

Topological Complexity and Predictability in the Dynamics of a Tumor Growth Model with Shilnikov's Chaos

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

Strong and weak Allee effects and chaotic dynamics in Richards' growths

In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.

Harvesting dynamic aspects from a real rational map : the bulb of period 5

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Um novo gerador central de padrões de locomoção para geração de trajectórias em tempo real de robôs quadrúpedes

Neste trabalho estuda-se a geração de trajectórias em tempo real de um robô quadrúpede. As trajectórias podem dividir-se em duas componentes: rítmica e discreta. A componente rítmica das trajectórias é modelada por uma rede de oito osciladores acoplados, com simetria 4 2 Z  Z . Cada oscilador é modelado matematicamente por um sistema de Equações Diferenciais Ordinárias. A referida rede foi proposta por Golubitsky, Stewart, Buono e Collins (1999, 2000), para gerar os passos locomotores de animais quadrúpedes. O trabalho constitui a primeira aplicação desta rede à geração de trajectórias de robôs quadrúpedes. A derivação deste modelo baseia-se na biologia, onde se crê que Geradores Centrais de Padrões de locomoção (CPGs), constituídos por redes neuronais, geram os ritmos associados aos passos locomotores dos animais. O modelo proposto gera soluções periódicas identificadas com os padrões locomotores quadrúpedes, como o andar, o saltar, o galopar, entre outros. A componente discreta das trajectórias dos robôs usa-se para ajustar a parte rítmica das trajectórias. Este tipo de abordagem é útil no controlo da locomoção em terrenos irregulares, em locomoção guiada (por exemplo, mover as pernas enquanto desempenha tarefas discretas para colocar as pernas em localizações específicas) e em percussão. Simulou-se numericamente o modelo de CPG usando o oscilador de Hopf para modelar a parte rítmica do movimento e um modelo inspirado no modelo VITE para modelar a parte discreta do movimento. Variou-se o parâmetro g e mediram-se a amplitude e a frequência das soluções periódicas identificadas com o passo locomotor quadrúpede Trot, para variação deste parâmetro. A parte discreta foi inserida na parte rítmica de duas formas distintas: (a) como um offset, (b) somada às equações que geram a parte rítmica. Os resultados obtidos para o caso (a), revelam que a amplitude e a frequência se mantêm constantes em função de g. Os resultados obtidos para o caso (b) revelam que a amplitude e a frequência aumentam até um determinado valor de g e depois diminuem à medida que o g aumenta, numa curva quase sinusoidal. A variação da amplitude das soluções periódicas traduz-se numa variação directamente proporcional na extensão do movimento do robô. A velocidade da locomoção do robô varia com a frequência das soluções periódicas, que são identificadas com passos locomotores quadrúpedes.

Robôs quadrúpedes: geração de trajectórias em tempo real usando sistemas dinâmicos não-lineares

A geração de trajectórias de robôs em tempo real é uma tarefa muito complexa, não existindo ainda um algoritmo que a permita resolver de forma eficaz. De facto, há controladores eficientes para trajectórias previamente definidas, todavia, a adaptação a variações imprevisíveis, como sendo terrenos irregulares ou obstáculos, constitui ainda um problema em aberto na geração de trajectórias em tempo real de robôs. Neste trabalho apresentam-se modelos de geradores centrais de padrões de locomoção (CPGs), inspirados na biologia, que geram os ritmos locomotores num robô quadrúpede. Os CPGs são modelados matematicamente por sistemas acoplados de células (ou neurónios), sendo a dinâmica de cada célula dada por um sistema de equações diferenciais ordinárias não lineares. Assume-se que as trajectórias dos robôs são constituídas por esta parte rítmica e por uma parte discreta. A parte discreta pode ser embebida na parte rítmica, (a.1) como um offset ou (a.2) adicionada às expressões rítmicas, ou (b) pode ser calculada independentemente e adicionada exactamente antes do envio dos sinais para as articulações do robô. A parte discreta permite inserir no passo locomotor uma perturbação, que poderá estar associada à locomoção em terrenos irregulares ou à existência de obstáculos na trajectória do robô. Para se proceder á análise do sistema com parte discreta, será variado o parâmetro g. O parâmetro g, presente nas equações da parte discreta, representa o offset do sinal após a inclusão da parte discreta. Revê-se a teoria de bifurcação e simetria que permite a classificação das soluções periódicas produzidas pelos modelos de CPGs com passos locomotores quadrúpedes. Nas simulações numéricas, usam-se as equações de Morris-Lecar e o oscilador de Hopf como modelos da dinâmica interna de cada célula para a parte rítmica. A parte discreta é modelada por um sistema inspirado no modelo VITE. Medem-se a amplitude e a frequência de dois passos locomotores para variação do parâmetro g, no intervalo [-5;5]. Consideram-se duas formas distintas de incluir a parte discreta na parte rítmica: (a) como um (a.1) offset ou (a.2) somada nas expressões que modelam a parte rítmica, e (b) somada ao sinal da parte rítmica antes de ser enviado às articulações do robô. No caso (a.1), considerando o oscilador de Hopf como dinâmica interna das células, verifica-se que a amplitude e frequência se mantêm constantes para -50.2. A extensão do movimento varia de forma directamente proporcional à amplitude. No caso das equações de Morris-Lecar, quando a componente discreta é embebida (a.2), a amplitude e a frequência aumentam e depois diminuem para - 0.170.5 Pode concluir-se que: (1) a melhor forma de inserção da parte discreta que menos perturbação insere no robô é a inserção como offset; (2) a inserção da parte discreta parece ser independente do sistema de equações diferenciais ordinárias que modelam a dinâmica interna de cada célula. Como trabalho futuro, é importante prosseguir o estudo das diferentes formas de inserção da parte discreta na parte rítmica do movimento, para que se possa gerar uma locomoção quadrúpede, robusta, flexível, com objectivos, em terrenos irregulares, modelada por correcções discretas aos padrões rítmicos.

Biaxial Nematic Order in the Hard-boomerang Fluid

We consider a fluid of hard boomerangs, each composed of two hard spherocylinders joined at their ends at an angle Psi. The resulting particle is nonconvex and biaxial. The occurence of nematic order in such a system has been investigated using Straley's theory, which is a simplificaton of Onsager's second-virial treatment of long hard rods, and by bifurcation analysis. The excluded volume of two hard boomerangs has been approximated by the sum of excluded volumes of pairs of constituent spherocylinders, and the angle-dependent second-virial coefficient has been replaced by a low-order interpolating function. At the so-called Landau point, Psi(Landau)approximate to 107.4 degrees, the fluid undergoes a continuous transition from the isotropic to a biaxial nematic (B) phase. For Psi not equal Psi(Landau) ordering is via a first-order transition into a rod-like uniaxial nematic phase (N(+)) if Psi > Psi(Landau), or a plate-like uniaxial nematic (N(-)) phase if Psi < Psi(Landau). The B phase is separated from the N(+) and N(-) phases by two lines of continuous transitions meeting at the Landau point. This topology of the phase diagram is in agreement with previous studies of spheroplatelets and biaxial ellipsoids. We have checked the accuracy of our theory by performing numerical calculations of the angle-dependent second virial coefficient, which yields Psi(Landau)approximate to 110 degrees for very long rods, and Psi(Landau)approximate to 90 degrees for short rods. In the latter case, the I-N transitions occur at unphysically high packing fractions, reflecting the inappropriateness of the second-virial approximation in this limit.

Synchronization and Basins of Synchronized in Two-Dimensional Piecewise Maps Via Coupling Three Pieces of One-Dimensional Maps

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

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.