A proposta deste trabalho é apresentar resultados para o estudo sistemático de sistemas dinâmicos reversíveis-equivariantes, ou seja, em presença simultânea de simetrias e antisimetrias. Este é o caso em que o domínio e as equações que regem o sistema são invariantes pela ação de um grupo de Lie compacto Γ formado pelas simetrias e anti-simetrias do problema. Apresentamos métodos de teoria de Singularidades e teoria de invariantes para classificar bifurcações a um parâmetro de pontos de equilíbrio destes sistemas. Para isso, separamos o estudo de aplicações Γ-reversíveis-equivariantes em dois casos: auto-dual e não auto-dual. No primeiro caso, a existência de um isomorfismo linear Γ-reversível-equivariante estabelece uma correspondência entre a classificação de problemas Γ-reversíveis-equivariantes e a classificação de problemas Γ-equivariantes associados, para os quais todos os elementos de Γ agem como simetria. Os resultados obtidos para o caso não auto-dual se baseiam em teoria de invariantes e envolvem técnicas algébricas que reduzem a análise ao caso polinomial invariante. Dois algoritmos simbólicos são estabelecidos para o cálculo de geradores para o módulo das funções anti-invariantes e para o módulo das aplicações reversíveis-equivariantes. / The purpose of this work is to present results for the sistematic study of reversible-equivariant dynamical systems, namely in simultaneous presence of symmetries and reversing simmetries. This is the case when the domain and the equations modeling the system are invariant under the action of a compact Lie group Γ formed by the symmetries and reversing symmetries of the problem. We present methods in Singularities and Invariant theory to classify oneparameter steady-state bifurcations of these systems. For that, we split the study of the ¡¡reversible-equivariant mapping into two cases: self-dual and non self-dual. In the first case, the existence of a Γ-reversible-equivariant linear isomorphism establishes a one-toone correspondence between the classification of Γ-reversible-equivariant problems and the classification of the associated Γ-equivariant problems, for which all elements in Γ act as symmetries. The results obtained for the non self-dual case are based on Invariant theory and involve algebraic techniques that reduce the analysis to the invariant polynomial case. Two symbolic algorithms are established for the computation of generators for the module of anti-invariant functions and for the module of reversible-equivariant mappings.

A proposta deste trabalho é apresentar resultados para o estudo sistemático de sistemas dinâmicos reversíveis-equivariantes, ou seja, em presença simultânea de simetrias e antisimetrias. Este é o caso em que o domínio e as equações que regem o sistema são invariantes pela ação de um grupo de Lie compacto Γ formado pelas simetrias e anti-simetrias do problema. Apresentamos métodos de teoria de Singularidades e teoria de invariantes para classificar bifurcações a um parâmetro de pontos de equilíbrio destes sistemas. Para isso, separamos o estudo de aplicações Γ-reversíveis-equivariantes em dois casos: auto-dual e não auto-dual. No primeiro caso, a existência de um isomorfismo linear Γ-reversível-equivariante estabelece uma correspondência entre a classificação de problemas Γ-reversíveis-equivariantes e a classificação de problemas Γ-equivariantes associados, para os quais todos os elementos de Γ agem como simetria. Os resultados obtidos para o caso não auto-dual se baseiam em teoria de invariantes e envolvem técnicas algébricas que reduzem a análise ao caso polinomial invariante. Dois algoritmos simbólicos são estabelecidos para o cálculo de geradores para o módulo das funções anti-invariantes e para o módulo das aplicações reversíveis-equivariantes. / The purpose of this work is to present results for the sistematic study of reversible-equivariant dynamical systems, namely in simultaneous presence of symmetries and reversing simmetries. This is the case when the domain and the equations modeling the system are invariant under the action of a compact Lie group Γ formed by the symmetries and reversing symmetries of the problem. We present methods in Singularities and Invariant theory to classify oneparameter steady-state bifurcations of these systems. For that, we split the study of the ¡¡reversible-equivariant mapping into two cases: self-dual and non self-dual. In the first case, the existence of a Γ-reversible-equivariant linear isomorphism establishes a one-toone correspondence between the classification of Γ-reversible-equivariant problems and the classification of the associated Γ-equivariant problems, for which all elements in Γ act as symmetries. The results obtained for the non self-dual case are based on Invariant theory and involve algebraic techniques that reduce the analysis to the invariant polynomial case. Two symbolic algorithms are established for the computation of generators for the module of anti-invariant functions and for the module of reversible-equivariant mappings.

A city is considered as a complex system. It consists of numerous interactivesub-systems and is affected by diverse factors including governmental landpolicies, population growth, transportation infrastructure, and market behavior.Land use and transportation systems are considered as the two most importantsubsystems determining urban form and structure in the long term. Meanwhile,urban growth is one of the most important topics in urban studies, and its maindriving forces are population growth and transportation development. Modelingand simulation are believed to be powerful tools to explore the mechanisms ofurban evolution and provide planning support in growth management. The overall objective of the thesis is to analyze and model urban growth basedon the simulation of land-use changes and the modeling of road networkexpansion. Since most previous urban growth models apply fixed transportnetworks, the evolution of road networks was particularly modeled. Besides,urban growth modeling is an interdisciplinary field, so this thesis made bigefforts to integrate knowledge and methods from other scientific and technicalareas to advance geographical information science, especially the aspects ofnetwork analysis and modeling. A multi-agent system was applied to model urban growth in Toronto whenpopulation growth is considered as being the main driving factor of urbangrowth. Agents were adopted to simulate different types of interactiveindividuals who promote urban expansion. The multi-agent model with spatiotemporalallocation criterions was shown effectiveness in simulation. Then, anurban growth model for long-term simulation was developed by integratingland-use development with procedural road network modeling. The dynamicidealized traffic flow estimated by the space syntax metric was not only used forselecting major roads, but also for calculating accessibility in land-usesimulation. The model was applied in the city centre of Stockholm andconfirmed the reciprocal influence between land use and street network duringthe long-term growth. To further study network growth modeling, a novel weighted network model,involving nonlinear growth and neighboring connections, was built from theperspective of promising complex networks. Both mathematical analysis andnumerical simulation were examined in the evolution process, and the effects ofneighboring connections were particular investigated to study the preferentialattachment mechanisms in the evolution. Since road network is a weightedplanar graph, the growth model for urban street networks was subsequentlymodeled. It succeeded in reproducing diverse patterns and each pattern wasexamined by a series of measures. The similarity between the properties of derived patterns and empirical studies implies that there is a universal growthmechanism in the evolution of urban morphology. To better understand the complicated relationship between land use and roadnetwork, centrality indices from different aspects were fully analyzed in a casestudy over Stockholm. The correlation coefficients between different land-usetypes and road network centralities suggest that various centrality indices,reflecting human activities in different ways, can capture land development andconsequently influence urban structure. The strength of this thesis lies in its interdisciplinary approaches to analyze andmodel urban growth. The integration of ‘bottom-up’ land-use simulation androad network growth model in urban growth simulation is the major contribution.The road network growth model in terms of complex network science is anothercontribution to advance spatial network modeling within the field of GIScience.The works in this thesis vary from a novel theoretical weighted network modelto the particular models of land use, urban street network and hybrid urbangrowth, and to the specific applications and statistical analysis in real cases.These models help to improve our understanding of urban growth phenomenaand urban morphological evolution through long-term simulations. Thesimulation results can further support urban planning and growth management.The study of hybrid models integrating methods and techniques frommultidisciplinary fields has attracted a lot attention and still needs constantefforts in near future. / <p>QC 20130514</p>