Dans cette thèse, nous étudions la robustesse dans le contexte de la modélisation de systèmes complexes par les automates cellulaires. En effet, si l'on cherche à reproduire un comportement émergent à partir d'un modèle d'automate cellulaire, il nous semble nécessaire de se demander si les comportements observés sont bien le résultat d'interactions entre entités constituantes, ou bien s'ils dépendent d'une définition particulière du modèle. Nous allons ainsi être amenés à considérer la robustesse du modèle, à savoir la résistance de son comportement à de petites variations sur les attributs de sa définition. Dans un premier temps, nous montrons la pertinence de cette approche en considérant plusieurs définitions possibles d'une perturbation de la mise à jour globale et en les appliquant à une classe simple et représentative de modèles d'automates cellulaires, les Automates Cellulaires Elémentaires. Nous observons que, malgré le fait que nos perturbations soient proches et qu'une majorité des modèles considérés ne change pas de comportement, quelques cas particuliers montrent des changements qualitatifs du comportement que nous étudions plus en détail. Dans un second temps, nous appliquons cette approche en nous penchant sur un modèle particulier d'automate cellulaire, qui simule le phénomène de formation d'essaim à partir d'un modèle évolué d'automate cellulaire, le gaz sur réseau. Nous explorons la robustesse du comportement du modèle en considérant la perturbation de deux attributs du modèle, la forme de la grille cellulaire et la mise à jour globale, et en tirons les conclusions sur la relation entre l'observation du comportement et la définition précise du modèle / In this thesis, we study the role of robustness in the context of the modelling of complex systems by cellular automata. Indeed, if we consider a cellular automaton which aims at reproducing an emergent behaviour from a similar structure, we want to determine whether its observed dynamics are the result of the interaction of entities, or whether it depends a precise definition of the model. We thus consider the model's robustness, that is, the resistance of the behaviour to small perturbations on the model features. First, we show the relevance of this approach by considering several definitions of a perturbation of the global updating and by applying them to a simple and representative class of cellular automata, the Elementary Cellular Automata. We observe that, despite the fact that most models show little or no change between the different perturbations, some particular cases show qualitative changes that we study in detail. Second, we apply this approach to a particular model of cellular automata, which simulates a swarming behaviour based on a lattice-gas model. We then explore the model robustness by considering the pertubations of two of the model's attributes, the lattice shape and the global updating, and discuss the relationship between the observation of the behaviour and the precise definitions of the model

Nos últimos anos, uma míriade de modelos de propagação de opinião foram propostos, motivados pelo interesse crescente dos físicos por problemas interdisciplinares tanto em sociologia, quanto em economia e biologia. Um dos objetivos desse trabalho é unificar alguns desses modelos em uma mesma formulação. Para isso, generalizamos a noção de confiança limitada para o que chamamos de regras de confiança, que podem ser interpretadas como a introdução de viéses ou preconceitos nas interações de agentes com opiniões distintas. Munidos dessa formulação, nos propusemos a estudar como modelos que promovem localmente conformidade (o que está de acordo com experimentos para grupos pequenos conduzidos por psicólogos), poderiam gerar diversidade globalmente (explicando a persistência de pontos de vista distintos em sociedades, por exemplo). Nós estudamos o campo médio do modelo do votante e de variantes do modelo Sznajd. Aplicando ferramentas de sistemas dinâmicos, conseguimos resolver analiticamente o comportamento qualitativo dos modelos na ausência de ruído e desenvolvemos uma teoria de perturbação para o modelo Sznajd com ruído infinitesimal, que nos forneceu um retrato parcial do comportamento na presença de ruído. Na ausência de ruído, chegamos a conclusão que o modelo do votante se comporta de maneira completamente diferente, enquanto que os outros modelos tem essencialmente o mesmo comportamento. Também fizemos simulações em redes Barabási-Albert e Watts-Strogatz para os modelos votante e Sznajd e, em colaboração com o grupo de pesquisa do Institute for Complex Systems and Mathematical Biology da Universidade de Aberdeen, estudamos um modelo de biodiversidade que pode ser encarado como uma variante do modelo do votante em uma rede quadrada. As nossas conclusões apontam que os resultados de campo médio podem ser compreendidos através de conexões com teoria de grafos e que os diversos modelos simulados se comportam em um certo sentido da mesma maneira, reforçando a idéia de universalidade entre eles (na verdade é essencial que existam aspectos universais no comportamento humano para que a modelagem de sistemas sociais seja factível, dadas as dificuldades óbvias de se construir um modelo realista para uma pessoa ou uma sociedade). Grosso modo, em todos os sistemas estudados, a coexistência ou não de pontos de vista diferentes parece depender mais crucialmente da rede e do tipo de regra de confiança, do que de outros detalhes específicos do modelo. / In the recent years, a great number of opinion propagation models were proposed, motivated by the increasing interest among physicists in interdisciplinary problems, not only in sociology, but also in economics and biology. One of the goals of this work is to unify some of these models under a same formulation. In order to do that, we generalized the notion of bounded confidence to what we called confidence rules, that can be interpreted as the introduction of biases and prejudices in the interactions among agents holding differing points of view. Using this formulation, we decided to study how models that locally breed conformity (what is in accordance with experiments conducted by psichologists for small groups) could sustain diversity globally (explaining the persistence of different points of view in societies, for example). We studied the mean field version of the voter model and of variants of the Sznajd model. We used dynamical systems techniques and were able to solve analytically the qualitative behaviour of the models in the absence of noise and developed a perturbation theory for the Sznajd model with infinitesimal noise, that yielded a partial picture of the behaviour with noise. In the absence of noise, we found that the voter model has a completely different behaviour, while the other models have essentially the same behaviour. We also did simulations in Barabási-Albert and Watts-Strogatz networks for the voter and the Sznajd models and we collaborated with the research group of the Institute for Complex Systems and Mathematical Biology from the University of Aberdeen, studying a biodiversity model that can be seen as a modification of the voter model in a square lattice. Our conclusions point that the mean field results can be understood through connections with graph theory problems and that the different models that were simulated, in some sense, have the same behaviour, reinforcing the idea of universality for these models (due to the obvious difficulties in modelling human beings in a reliable and realistic way, some degree of universality in human behaviour is actually essential, in order for social modelling to be feasible). Roughly speaking, in all the systems that were studied, the coexistence or not of differing opinions, seems to depend more strongly on the network and on the type of confidence rule used, than in other specific details of the model.

Uncertainties in both loading and structural characteristics can adversely affect the response and reliability of a structure. Parameter uncertainties in structural dynamics can arise due to several sources. These include variations due to intrinsic material property variability, measurement errors, manufacturing and assembly errors, differences in modeling and solution procedures. Problems of structural dynamics with randomly distributed spatial inhomogeneities in elastic, mass, and damping properties, have been receiving wide attention. Several mathematical and computational issues include discretization of random fields, characterization of random eigensolutions, inversion of random matrices, solutions of stochastic boundary-value problems, and description of random matrix products. Difficulties are encountered when one has to include interaction between nonlinear and stochastic system characteristics, or if one is interested in controlling the system response. The study of structural systems including the effects of system nonlinearity in the presence of parameter uncertainties presents serious challenges and difficulties to designers and reliability engineers. In the analysis of large structures, the situation for substructuring frequently arises due to the repetition of identical assemblages (substructures), within a structure, and the general need to reduce the size of the problem, particularly in the case of non-linear inelastic dynamic analysis. A small reduction in the model size can have a large effect on the storage and time requirement. A primary structural dynamic system may be coupled to subsystems such as piping systems in a nuclear reactor or in a chemical plant. Usually subsystem in itself is quite complex and its modeling with finite elements may result in a large number of degrees of freedom. The reduced subsystem model should be of low-order yet capturing the essential dynamics of the subsystem for useful integration with the primary structure. There are two major issues to be studied: one, techniques for analyzing a complex structure into component subsystems, analyzing the individual sub-system dynamics, and from thereon determining the dynamics of the structure after assembling the subsystems. The nonlinearity due to support gap effects such as supports for piping system in nuclear reactors further complicates the problem. The second is the issue of reviewing the methods for reducing the model-order of the component subsystems such that the order of the global dynamics, after assembly, is within some predefined limits. In the reliability analysis of complex engineering structures, a very large number of the system parameters have to be considered as random variables. The parameter uncertainties are modeled as random variables and are assumed to be time independent. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. The procedure involves the reduction of the size of the vector of random variables before the calculation of failure probability. The objectives of this thesis are: 1.To use the available model reduction techniques in order to effectively reduce the size of the finite element model, and hence, compare the dynamic responses from such models. 2.Study of propagation of uncertainties in the reduced order/coupled stochastic finite element dynamic models. 3.Addressing the localized nonlinearities due to support gap effects in the built up structures, and also in cases of sudden change in soil behaviour under the footings. The irregularity in soil behaviour due to lateral escape of soil due to failure of quay walls/retaining walls/excavation in neighbouring site, etc. 4.To evolve a procedure for the reduction of size of the vector containing the random variables before the calculation of failure probability. In the reliability analysis of complex engineering structures, a very large number of the system parameters are considered to be random variables. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. 5.To analyze the reduced nonlinear stochastic dynamic system (with phase space reduction), and effectively using the network pruning technique for the solution, and also to use filter theory (wavelet theory) for reducing the input earthquake record to save computational time and cost. It is believed that the techniques described provide highly useful insights into the manner structural uncertainties propagate. The cross-sectional area, length, modulus of elasticity and mass density of the structural components are assumed as random variables. Since both the random and design variables are expressed in a discretized parameter space, the stochastic sensitivity function can be modeled in a parallel way. The response of the structures in frequency domain is considered. This thesis is organized into seven chapters. This thesis deals with the reduced order models of the stochastic structural systems under deterministic/random loads. The Chapter 1 consists of a brief introduction to the field of study. In Chapter 2, an extensive literature survey based on the previous works on model order reduction and the response variability of the structural dynamic systems is presented. The discussion on parameter uncertainties, stochastic finite element method, and reliability analysis of structures is covered. The importance of reducing mechanical models for dynamic response variability, the systems with high-dimensional variables and reduction in random variables space, nonlinearity issues are discussed. The next few chapters from Chapter 3 to Chapter 6 are the main contributions in this thesis, on model reduction under various situations for both linear and nonlinear systems. After forming a framework for model reduction, local nonlinearities like support gaps in structural elements are considered. Next, the effect of reduction in number of random variables is tackled. Finally influence of network pruning and decomposition of input signals into low and high frequency parts are investigated. The details are as under. In Chapter 3, the issue of finite element model reduction is looked into. The generalized finite element analysis of the full model of a randomly parametered structure is carried out under a harmonic input. Different well accepted finite element model reduction techniques are used for FE model reduction in the stochastic dynamic system. The structural parameters like, mass density and modulus of elasticity of the structural elements are considered to be non-Gaussian random variables. Since the variables considered here are strictly positive, the probabilistic distribution of the random variables is assumed to be lognormal. The sensitivities in the eigen solutions are compared. The response statistics based on response of models in frequency domain are compared. The dynamic responses of the full FE model, separated into real and imaginary parts, are statistically compared with those from reduced FE models. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 4, the problem of coupling of substructures in a large and complex structure, and FE model reduction, e.g., component mode synthesis (CMS) is studied in the stochastic environment. Here again, the statistics of the response from full model and reduced models are compared. The issues of non-proportional damping, support gap effects and/local nonlinearity are considered in the stochastic sense. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 5, the reduction in size of the vector of random variables in the reliability analysis is attempted. Here, the relative entropy/ K-L divergence/mutual information, between the random variables is considered as a measure for ranking of random variables to study the influence of each random variable on the response/reliability of the structure. The probabilistic distribution of the random variables is considered to be lognormal. The reliability analysis is carried out with the well known Bucher and Bourgund algorithm (1990), along with the probabilistic model reduction of the stochastic structural dynamic systems, within the framework of response surface method. The reduction in number of random variables reduces the computational effort required to construct an approximate closed form expression in response surface approach. In Chapter 6, issues regarding the nonlinearity effects in the reduced stochastic structural dynamic systems (with phase space reduction), along with network pruning are attempted. The network pruning is also adopted for reduction in computational effort. The earthquake accelerogram is decomposed using Fast Mallat Algorithm (Wavelet theory) into smaller number of points and the dynamic analysis of structures is carried out against these reduced points, effectively reducing the computational time and cost. Chapter 7 outlines the contributions made in this thesis, together with a few suggestions made for further research. All the finite element codes were developed using MATLAB5.3. Final pages of the thesis contain the references made in the preparation of this thesis.

The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that fractional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calculus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the information about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or instability. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability, it is important to study both the top Lyapunov exponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the Stratonovich equations of motion are set up, and then decoupled into four-dimensional Ito stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Ito equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations.

Nos últimos anos, uma míriade de modelos de propagação de opinião foram propostos, motivados pelo interesse crescente dos físicos por problemas interdisciplinares tanto em sociologia, quanto em economia e biologia. Um dos objetivos desse trabalho é unificar alguns desses modelos em uma mesma formulação. Para isso, generalizamos a noção de confiança limitada para o que chamamos de regras de confiança, que podem ser interpretadas como a introdução de viéses ou preconceitos nas interações de agentes com opiniões distintas. Munidos dessa formulação, nos propusemos a estudar como modelos que promovem localmente conformidade (o que está de acordo com experimentos para grupos pequenos conduzidos por psicólogos), poderiam gerar diversidade globalmente (explicando a persistência de pontos de vista distintos em sociedades, por exemplo). Nós estudamos o campo médio do modelo do votante e de variantes do modelo Sznajd. Aplicando ferramentas de sistemas dinâmicos, conseguimos resolver analiticamente o comportamento qualitativo dos modelos na ausência de ruído e desenvolvemos uma teoria de perturbação para o modelo Sznajd com ruído infinitesimal, que nos forneceu um retrato parcial do comportamento na presença de ruído. Na ausência de ruído, chegamos a conclusão que o modelo do votante se comporta de maneira completamente diferente, enquanto que os outros modelos tem essencialmente o mesmo comportamento. Também fizemos simulações em redes Barabási-Albert e Watts-Strogatz para os modelos votante e Sznajd e, em colaboração com o grupo de pesquisa do Institute for Complex Systems and Mathematical Biology da Universidade de Aberdeen, estudamos um modelo de biodiversidade que pode ser encarado como uma variante do modelo do votante em uma rede quadrada. As nossas conclusões apontam que os resultados de campo médio podem ser compreendidos através de conexões com teoria de grafos e que os diversos modelos simulados se comportam em um certo sentido da mesma maneira, reforçando a idéia de universalidade entre eles (na verdade é essencial que existam aspectos universais no comportamento humano para que a modelagem de sistemas sociais seja factível, dadas as dificuldades óbvias de se construir um modelo realista para uma pessoa ou uma sociedade). Grosso modo, em todos os sistemas estudados, a coexistência ou não de pontos de vista diferentes parece depender mais crucialmente da rede e do tipo de regra de confiança, do que de outros detalhes específicos do modelo. / In the recent years, a great number of opinion propagation models were proposed, motivated by the increasing interest among physicists in interdisciplinary problems, not only in sociology, but also in economics and biology. One of the goals of this work is to unify some of these models under a same formulation. In order to do that, we generalized the notion of bounded confidence to what we called confidence rules, that can be interpreted as the introduction of biases and prejudices in the interactions among agents holding differing points of view. Using this formulation, we decided to study how models that locally breed conformity (what is in accordance with experiments conducted by psichologists for small groups) could sustain diversity globally (explaining the persistence of different points of view in societies, for example). We studied the mean field version of the voter model and of variants of the Sznajd model. We used dynamical systems techniques and were able to solve analytically the qualitative behaviour of the models in the absence of noise and developed a perturbation theory for the Sznajd model with infinitesimal noise, that yielded a partial picture of the behaviour with noise. In the absence of noise, we found that the voter model has a completely different behaviour, while the other models have essentially the same behaviour. We also did simulations in Barabási-Albert and Watts-Strogatz networks for the voter and the Sznajd models and we collaborated with the research group of the Institute for Complex Systems and Mathematical Biology from the University of Aberdeen, studying a biodiversity model that can be seen as a modification of the voter model in a square lattice. Our conclusions point that the mean field results can be understood through connections with graph theory problems and that the different models that were simulated, in some sense, have the same behaviour, reinforcing the idea of universality for these models (due to the obvious difficulties in modelling human beings in a reliable and realistic way, some degree of universality in human behaviour is actually essential, in order for social modelling to be feasible). Roughly speaking, in all the systems that were studied, the coexistence or not of differing opinions, seems to depend more strongly on the network and on the type of confidence rule used, than in other specific details of the model.

No description available.

Cette thèse porte sur la dynamique, normale à l'interface, d'un contact sec en glissement stationnaire entre deux surfaces de topographies aléatoires, soumis à une faible charge normale. Dans ce contexte, le mouvement d'un patin sous son propre poids a été étudié expérimentalement. Des mesures par vibrométrie laser du déplacement et de la vitesse normale du patin ont confirmé que, lorsque la vitesse de glissement augmente, le patin transite entre un régime où le contact est permanent vers un régime dynamique où il subit décollements, chocs et rebonds.À basse vitesse, le mouvement normal résulte d'un filtrage géométrique des topographies. Les caractéristiques statistiques et spectrales de ce mouvement ont pu être décrites. Les influences de la rugosité, de la longueur de corrélation, de la largeur de bande du spectre de rugosité et de l'aire apparente de contact ont été identifiées et analysées. Ces résultats ont pu être reproduits par des modèles numériques, mais aussi analytiques en adaptant la théorie des valeurs extrêmes. Des modèles de type Bouncing Ball, dont l'excitation est supposée donnée par le processus de filtrage géométrique, ont également été mis en place. Ils reproduisent une large gamme d'observations en régime dynamique, de la transition aux vibro-impacts.Pour tester certaines hypothèses des modèles mis en place, un patin multi-voies original a été développé et a permis d'accéder à la localisation spatiale des micro-contacts transitoires entre surfaces antagonistes. On observe que les micro-contacts sont gouvernés par une longueur caractéristique à basse vitesse de glissement et par un temps caractéristique à haute vitesse. Les rotations du patin deviennent importantes à haute vitesse, modifiant la répartition des micro-contacts à la surface du patin. / This PhD thesis addresses the issue of the dynamics, normal to the interface, of a dry steady-sliding contact between two random topographies under weak normal load. In this context, the motion of a slider under its own weight has been studied experimentally. Measurements, using a laser vibrometer, of the normal displacement and velocity of the slider confirm the existence of a transition, as the sliding speed increases, from a regime of permanent contact to a regime of lift-offs, shocks and rebounds.At low speed, the normal motion is due to a geometrical filtering of the topographies, the statistical and spectral properties of which have been described. The roles of the roughness, including its spectral breadth and correlation length, and of the apparent contact area have been identified and analyzed. Those results have been reproduced not only using numerical models, but also using analytical ones based on the extreme value theory. Bouncing-Ball-like models, the excitation of which is assumed to be given by the geometrical filtering, have also been implemented and match with a broad range of experimental observations in dynamical regime, from the transition to vibro-impacts.To test some of the hypothesis of the models, a new experimental multi-channel slider has been designed and has enabled access to the spatial localization of the transient micro-contacts between the antagonists surfaces. It has been shown that micro-contacts are governed by a characteristic length at low sliding speed and by a characteristic time at high speed. The rotational motion of the slider also increases with sliding speed, changing the micro-contact distribution along the surface of the slider.

No description available.

[pt] É um resultado conhecido na literatura que uma cadeia unidimensional de partículas, que interagem harmonicamente com seus primeiros vizinhos, não conduz calor, e forças não lineares são necessárias para reproduzir a lei de Fourier da condução de calor. Quando são introduzidas assimetrias em tal sistema condutor, se obtém um efeito retificador onde a corrente térmica apresenta magnitudes diferentes dependendo de qual lado da cadeia tem maior temperatura, tais dispositivos sendo chamados de diodos térmicos. Neste trabalho estudamos os dois fenômenos, condução de calor e retificação térmica, em uma cadeia unidimensional de partículas, com condições de contorno fixas, acopladas a dois banhos térmicos, um em cada extremidade, modelados como termostatos de Langevin. As partículas interagem com seus primeiros vizinhos harmonicamente e estão sujeitas a um potencial localizado externo não linear, para o qual estudamos dois tipos, os potenciais Frenkel-Kontorova e Ø elevado a 4. Verificamos que a lei de Fourier é observada, para ambos os casos, com o perfil de temperatura e a condutividade térmica dependendo da relação entre as amplitudes harmônica e anarmônica, e a temperatura média do sistema. Em seguida, para criar uma assimetria na cadeia, nós acoplamos dois segmentos de mesmo tamanho. Observamos um efeito retificador onde a direção preferencial difere para cada potencial localizado estudado. A forma como as temperaturas dos banhos térmicos mudam a magnitude da retificação também foi observada. Nós também investigamos o efeito de não linearidades interfaciais, por meio de uma lei de potência que acopla segmentos Ø elevado a 4. Alterando o expoente da lei de potência, nós buscamos as condições sob as quais a retificação ótima é atingida. / [en] It is a known result in the literature that a one-dimensional chain of particles that interact harmonically with its first neighbors does not conduct heat, and nonlinear forces are needed to reproduce Fourier s law of heat conduction. When asymmetries are introduced in such a conducting system, a rectifying effect is obtained where the thermal current shows different magnitudes depending on which side of the chain has higher temperature, such devices being called thermal diodes. In this work we study both phenomena, heat conduction and thermal rectification, in a onedimensional chain of particles, with fixed boundary conditions, coupled to two thermal baths, one at each end, modeled as Langevin thermostats. The particles interact with their first neighbors harmonically and have a nonlinear on-site potential, for which we study two types, Frenkel-Kontorova and Ø 4 potentials. We verify that, for both cases, Fourier s law is observed, where the temperature profile and the thermal conductivity are dependent on the relation between the harmonic and anharmonic amplitudes, and the system s average temperature. Next, to create an asymmetry in the chain, we coupled two different segments of equal lengths. We observed a rectifying effect, where the preferential direction differs for each of the two on-site potentials studied. How the heat-bath temperatures changes the magnitude of rectification was also observed. We also investigated the effect of interfacial nonlinearities through a power-law potential, coupling Ø 4 segments. By changing the power-law exponent, we looked for the conditions under which optimal rectification is achieved.

[pt] A dinâmica da distribuição da riqueza para o modelo conhecido em inglês como Yard-Sale Model (Modelo da Venda de Quintal) pode ser descrita através de uma equação de Fokker-Planck para a função densidade de probabilidade P(w, t) da riqueza w em um instante t. Neste trabalho foi investigado o efeito de um arrasto redistributivo não linear nessa dinâmica. Considera-se (I) uma taxação do tipo linear por partes, onde apenas aqueles com riqueza acima de um determinado valor são taxados, e, (II) uma taxação na forma de lei de potência, que inclui os tipos progressivo e regressivo. Em todos os casos, o total arrecadado é distribuído igualmente. Analisou-se como essas regras podem modificar a distribuição da riqueza numa população e, principalmente, o nível de desigualdade medido pelo índice de Gini. / [en] The dynamics of wealth distribution for the so-called Yard-Sale Model can be described by a Fokker-Planck equation for the probability density function P(w, t) of wealth w at time t. In this work, the effect of nonlinear redistributive drifts was investigated. It was considered (I) a piecewise linear tax, where only those with wealth above a certain threshold are taxed, and, (II) a power-law tax that includes the progressive and regressive types. In all cases, the collected amount of wealth is redistributed equally. We analyze how these rules modify the distribution of wealth across the population and, mainly, the inequality level measured through the Gini index.