Branching processes and computational collapse of discretized unimodal mappings

In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.

Mixed gas equilibrium adsorption on zeolites and energetic heterogeneity of adsorption volume

A model for binary mixture adsorption accounting for energetic heterogeneity and intermolecular interactions is proposed in this paper. The model is based on statistical thermodynamics, and it is able to describe molecular rearrangement of a mixture in a nonuniform adsorption field inside a cavity. The Helmholtz free energy obtained in the framework of this approach has upper and lower limits, which define a permissible range in which all possible solutions will be found. One limit corresponds to a completely chaotic distribution of molecules within a cavity, while the other corresponds to a maximum ordered molecular structure. Comparison of the nearly ideal O-2-N-2-zeolite NaX system at ambient temperature with the system Of O-2-N-2-zeolite CaX at 144 K has shown that a decrease of temperature leads to a molecular rearrangement in the cavity volume, which results from the difference in the fluid-solid interactions. The model is able to describe this behavior and therefore allows predicting mixture adsorption more accurately compared to those assuming energetic uniformity of the adsorption volume. Another feature of the model is its ability to correctly describe the negative deviations from Raoult's law exhibited by the O-2-N-2-CaX system at 144 K. Analysis of the highly nonideal CO2-C2H6-zeolite NaX system has shown that the spatial molecular rearrangement in separate cavities is induced by not only the ion-quadrupole interaction of the CO2 molecule but also the significant difference in molecular size and the difference between the intermolecular interactions of molecules of the same species and those of molecules of different species. This leads to the highly ordered structure of this system.

Quantum dynamical characterization of unimolecular resonances

We give a selective review of quantum mechanical methods for calculating and characterizing resonances in small molecular systems, with an emphasis on recent progress in Chebyshev and Lanczos iterative methods. Two archetypal molecular systems are discussed: isolated resonances in HCO, which exhibit regular mode and state specificity, and overlapping resonances in strongly bound HO2, which exhibit irregular and chaotic behavior. Future directions in this field are also discussed.

PACOTE ANTI-CORRUPÇÃO DE DILMA?

Olhemos para uma folha de São Paulo no meio do trânsito caótico e dos milhares de arranha-céus, enquanto passam centenas de buzinantes motoqueiros suicidas a uma velocidade estonteante… Depois de milhões de pessoas se terem manifestado no Brasil por causa de megas-processos de corrupção que têm sido descobertos, entre os quais o Petrolão, no qual as investigações apontam que a propina (suborno) foi o triplo do caso Mensalão, eis que a Presidenta Dilma avança para um pacote anti-corrupção. A operação Lava Jato já fala em milhares de milhões de reais. Mas, importante, é que apenas uma medida tem efeitos de aplicação imediata! São 7 as medidas. § Let us look for a Folha de Sao Paulo in the middle of chaotic traffic and thousands of skyscrapers, as they pass hundreds of suicide motorcycle at breakneck speed ... After millions of people have expressed in Brazil because of corruption mega-processes that have been discovered, including the Petrolão in which investigations indicate that the bribe (bribery) was triple the Mensalão case, behold, the President Dilma advances to an anti-corruption package. Operation Lava Jet is already talking about billions of reais. But, importantly, it is that only a measure takes effect immediately applicable! There are 7 steps.

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.

Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K, C-0) and (K, sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K-c decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

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.

Dynamical behaviour on the parameter space: new populational growth models proportional to beta densities

We present new populational growth models, generalized logistic models which are proportional to beta densities with shape parameters p and 2, where p > 1, with Malthusian parameter r. The complex dynamical behaviour of these models is investigated in the parameter space (r, p), in terms of topological entropy, using explicit methods, when the Malthusian parameter r increases. This parameter space is split into different regions, according to the chaotic behaviour of the models.

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.

Leis de potência em sistemas naturais e artificiais

A Lei de Potência é uma particularidade de um sistema não linear, revelando um sistema complexo próximo da auto-organização. Algumas características de sistemas naturais e artificiais, tais como dimensão populacional das cidades, valor dos rendimentos pessoais, frequência de ocorrência de palavras em textos e magnitude de sismos, seguem distribuições do tipo Lei de Potência. Estas distribuições indicam que pequenas ocorrências são muito comuns e grandes ocorrências são raras, podendo porém verificar-se com razoável probabilidade. A finalidade deste trabalho visa a identificação de fenómenos associados às Leis de Potência. Mostra-se o comportamento típico destes fenómenos, com os dados retirados dos vários casos de estudo e com a ajuda de uma meta-análise. As Leis de Potência em sistemas naturais e artificiais apresentam uma proximidade a um padrão, quando os valores são normalizados (frequências relativas) para dar origem a um meta-gráfico.

Modelação do transporte e dispersão de um marcador corado no Rio Febros

O rio Febros é um pequeno curso de água, situado no concelho de Vila Nova de Gaia, com cerca de 15 km de extensão, cuja bacia hidrográfica ocupa uma área de aproximadamente 35,4 km2. Nasce em Seixezelo e desagua na margem esquerda do Rio Douro no Cais do Esteiro, em Avintes. Em Maio de 2008, um acidente de viação teve como consequência o derrame de cerca de quatro toneladas de ácido clorídrico que rapidamente convergiu às águas do rio. Apenas um dia depois, o pH desceu para três e muitos foram os peixes que morreram. A solução adoptada para evitar o desaire foi introduzir milhares de litros de água de modo a diluir o ácido presente, ao longo de todo o curso de água. Tal facto não evitou a destruição de parte de um ecossistema, que ainda nos dias de hoje se encontra em recuperação. De forma a avaliar-se o impacto destas possíveis perturbações sejam estas de origem antropogénica ou natural é necessário possuir conhecimentos dos processos químicos tais como a advecção, a mistura devida à dispersão e a transferência de massa ar/água. Estes processos irão determinar o movimento e destino das substâncias que poderão ser descarregadas no rio. Para tal, recorrer-se-á ao estudo hidrogeométrico do curso de água assim como ao estudo do comportamento de um marcador, simulando uma possível descarga. A rodamina WT será o marcador a ser utilizado devido à panóplia de características ambientalmente favoráveis. Os estudos de campo com este corante, realizados em sequência de descarga previamente estudada, fornecem uma das melhores fontes de informação para verificação e validação de modelos hidráulicos utilizados em estudos de qualidade de águas e protecção ambiental. Escolheram-se dois pontos de descarga no Febros, um em Casal Drijo e outro no Parque Biológico de Gaia, possuindo cada um deles, a jusante, duas estações de monitorização. Pelo modelo ADE os valores obtidos para o coeficiente de dispersão longitudinal para as estações Pontão d’ Alheira, Pinheiral, Menesas e Giestas foram, respectivamente, 0,3622; 0,5468; 1,6832 e 1,7504 m2/s. Para a mesma sequência de estações, os valores da velocidade de escoamento obtidos neste trabalho experimental foram de 0,0633; 0,0684; 0,1548 e 0,1645 m/s. Quanto ao modelo TS, os valores obtidos para o coeficiente de dispersão longitudinal para as estações Pontão d’ Alheira, Pinheiral, Menesas e Giestas foram, respectivamente, 0,2339; 0,1618; 0,5057e 1,1320 m2/s. Para a mesma sequência de estações, os valores da velocidade de escoamento obtidos neste trabalho experimental foram de 0,0652; 0,0775; 0,1891 e 0,1676 m/s. Os resultados foram ajustados por um método directo, o método dos momentos, e por dois métodos indirectos, os modelos ADE e TS. O melhor ajuste corresponde ao modelo TS onde os valores do coeficiente de dispersão longitudinal e da velocidade de escoamento são aqueles que melhor se aproximam da realidade. Quanto ao método dos momentos, o valor estimado para a velocidade é de 0,162 m/s e para o coeficiente de dispersão longitudinal de 9,769 m2/s. Não obstante, a compreensão da hidrodinâmica do rio e das suas características, bem como a adequação de modelos matemáticos no tratamento de resultados formam uma estratégia de protecção ambiental inerente a futuros impactos que possam suceder.

Complete synchronization and delayed synchronization in couplings

We consider a general coupling of two identical chaotic dynamical systems, and we obtain the conditions for synchronization. We consider two types of synchronization: complete synchronization and delayed synchronization. Then, we consider four different couplings having different behaviors regarding their ability to synchronize either completely or with delay: Symmetric Linear Coupled System, Commanded Linear Coupled System, Commanded Coupled System with delay and symmetric coupled system with delay. The values of the coupling strength for which a coupling synchronizes define its Window of synchronization. We obtain analytically the Windows of complete synchronization, and we apply it for the considered couplings that admit complete synchronization. We also obtain analytically the Window of chaotic delayed synchronization for the only considered coupling that admits a chaotic delayed synchronization, the commanded coupled system with delay. At last, we use four different free chaotic dynamics (based in tent map, logistic map, three-piecewise linear map and cubic-like map) in order to observe numerically the analytically predicted windows.

Analysis of stock market indices with multidimensional scaling and wavelets

Stock market indices SMIs are important measures of financial and economical performance. Considerable research efforts during the last years demonstrated that these signals have a chaotic nature and require sophisticated mathematical tools for analyzing their characteristics. Classical methods, such as the Fourier transform, reveal considerable limitations in discriminating different periods of time. This paper studies the dynamics of SMI by combining the wavelet transform and the multidimensional scaling MDS . Six continuous wavelets are tested for analyzing the information content of the stock signals. In a first phase, the real Shannon wavelet is adopted for performing the evaluation of the SMI dynamics, while their comparison is visualized by means of the MDS. In a second phase, the other wavelets are also tested, and the corresponding MDS plots are analyzed.

A fractional approach to the Fermi-Pasta-Ulam problem

This paper studies the Fermi-Pasta-Ulam problem having in mind the generalization provided by Fractional Calculus (FC). The study starts by addressing the classical formulation, based on the standard integer order differential calculus and evaluates the time and frequency responses. A first generalization to be investigated consists in the direct replacement of the springs by fractional elements of the dissipative type. It is observed that the responses settle rapidly and no relevant phenomena occur. A second approach consists of replacing the springs by a blend of energy extracting and energy inserting elements of symmetrical fractional order with amplitude modulated by quadratic terms. The numerical results reveal a response close to chaotic behaviour.

A multi-objective approach for the motion planning of redundant manipulators

Kinematic redundancy occurs when a manipulator possesses more degrees of freedom than those required to execute a given task. Several kinematic techniques for redundant manipulators control the gripper through the pseudo-inverse of the Jacobian, but lead to a kind of chaotic inner motion with unpredictable arm configurations. Such algorithms are not easy to adapt to optimization schemes and, moreover, often there are multiple optimization objectives that can conflict between them. Unlike single optimization, where one attempts to find the best solution, in multi-objective optimization there is no single solution that is optimum with respect to all indices. Therefore, trajectory planning of redundant robots remains an important area of research and more efficient optimization algorithms are needed. This paper presents a new technique to solve the inverse kinematics of redundant manipulators, using a multi-objective genetic algorithm. This scheme combines the closed-loop pseudo-inverse method with a multi-objective genetic algorithm to control the joint positions. Simulations for manipulators with three or four rotational joints, considering the optimization of two objectives in a workspace without and with obstacles are developed. The results reveal that it is possible to choose several solutions from the Pareto optimal front according to the importance of each individual objective.