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Unidade:
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Instituto de Ciências Matemáticas e de Computação |
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Modalidade:
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Difusão |
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Tipo:
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Presencial |
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Público Alvo:
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Alunos de graduação, pós graduação, pos-doc, pesquisadores, publico em geral interessado em EDOs e aplicações |
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Objetivo:
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Many natural phenomena coming from the physics, chemist, economy, biology, can be modelized using polynomial differential systems in the plane. The main objective of the qualitative theory for a polynomial differential system is to draw its phase portrait, that is to describe the plane as union of all the orbits of the polynomial differential system. But since the plane is an open manifold it is important to control how the orbits come from or go to its boundary, i.e. how the orbits come from or go to the infinity of the plane. This is solved using the so-called Poincaré compactification. Doing this compactification the dynamics of the polynomial differential system is extended to a 2-dimensional sphere. Then the Poincaré-Hopf Theorem on this sphere helps to control the equilibrium points of the polynomial differential system.
The three main objectives are: first to introduce the Poincaré compactification of a polynomial differential system in the plane, second to prove the Poincaré-Hopf Theorem, and third to show how to use these two tools for describing the global dynamics of any polynomial differential system in the plane.
The Poincaré compactification is the tool which allow to describe how the orbits of a planar polynomial differential system come from or go to infinity. Without this tool is not possible to understand the global dynamics of the polynomial differential systems.
The Poincaré-Hopf Theorem is one of the strong theorems in mathematics, because it relates the local notion of the equilibrium points, with the global notion of the Euler characteristic of a manifold |
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Pré-requisito Graduação:
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Não |
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Área de Conhecimento:
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Matemática
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