Loading...
Publication
Sistemas dinâmicos não-lineares e aplicações
| Name: | Description: | Size: | Format: | |
|---|---|---|---|---|
| 5.61 MB | Adobe PDF |
Authors
Advisor(s)
Abstract(s)
Uma equação que contém uma, ou mais, das suas derivadas e uma função
desconhecida é chamada de equação diferencial. Dado isto, os objetivos desta dissertação
são estudar sistemas dinâmicos não-lineares, determinar a estabilidade de estados
estacionários, estudar o comportamento das órbitas no diagrama de fases, linearizar
sistemas e resolver problemas aplicados.
As equações diferenciais ordinárias de primeira ordem podem ser classificadas
conforme os métodos empregues na obtenção da sua solução explícita. As propriedades
qualitativas dos pontos de equilíbrio e, por sua vez, a sua estabilidade, podem ser obtidas
através do esboço do campo vetorial associado, sendo classificados por poço, ponto de
sela ou fonte. Nas equações diferenciais não-lineares autónomas de segunda ordem, o
seu estudo é, em geral, descrito no diagrama de fases. Introduzindo uma mudança de
variável, obtém-se um sistema diferencial não-linear de primeira ordem, onde a sucessão
de estados para os diversos valores da variável independente traçam as órbitas no
diagrama de fases. Os pontos de equilíbrio, neste diagrama, são classificados como
centros ou pontos de sela.
Nossistemas autónomos de primeira ordem, à medida que a variável independente
𝑡 varia, os pares (𝑥(𝑡), 𝑦(𝑡)) traçam um caminho de fase direcionado no plano 𝑥𝑂𝑦. Os
pontos de equilíbrio são classificados conforme o comportamento das órbitas à sua volta.
Quando não se consegue determinar a solução analítica nem a equação das órbitas no
diagrama de fases, recorre-se à técnica de linearização. Esta técnica transforma,
localmente, as propriedades um sistema não-linear num sistema linear. Quanto à
estabilidade dos pontos de equilíbrio e construção do respetivo diagrama de fases
associado, estes dependem dos valores próprios da matriz jacobiana do sistema linear,
podendo representar nós, pontos de sela, centros ou espirais.
An equation that contains one, or more, of its derivatives and an unknown function is called a differential equation. Given this, the objectives of this dissertation are to study nonlinear dynamic systems, to determine the stability of stationary states, to study the behavior of the orbits in the phase diagram, to linearize systems and to solve applied problems. First order ordinary differential equations can be classified according to the methods used to obtain their explicit solution. The qualitative properties of the equilibrium points and their stability, can be obtained by sketching the associated vector field, being classified by sink, saddle point or source. In second order autonomous nonlinear differential equations, their study is, in general, described in the phase diagram. By change of variables, a first order nonlinear differential system is obtained, such that the succession of states for the various values of the independent variable gives the orbits in the phase diagram. The equilibrium points in this diagram are classified as centers or saddle points. In first order autonomous systems, as the independent variable 𝑡 varies, the pairs (𝑥(𝑡), 𝑦(𝑡)) gives a phase path ordered in the 𝑥𝑂𝑦 plane. The equilibrium points are classified according to the behavior of the orbits in its neighborhood. When it is not possible to determine the analytical solution or the equation of the orbits in the phase diagram, the technique of linearization is used. This technique transforms, locally, the properties of a non-linear system into a linear system. As regards the stability of the equilibrium points and the construction of the respective associated phase diagram, they depend on the eigenvalues of the jacobian matrix of the linear system, which can represent nodes, saddle points, centers or spirals.
An equation that contains one, or more, of its derivatives and an unknown function is called a differential equation. Given this, the objectives of this dissertation are to study nonlinear dynamic systems, to determine the stability of stationary states, to study the behavior of the orbits in the phase diagram, to linearize systems and to solve applied problems. First order ordinary differential equations can be classified according to the methods used to obtain their explicit solution. The qualitative properties of the equilibrium points and their stability, can be obtained by sketching the associated vector field, being classified by sink, saddle point or source. In second order autonomous nonlinear differential equations, their study is, in general, described in the phase diagram. By change of variables, a first order nonlinear differential system is obtained, such that the succession of states for the various values of the independent variable gives the orbits in the phase diagram. The equilibrium points in this diagram are classified as centers or saddle points. In first order autonomous systems, as the independent variable 𝑡 varies, the pairs (𝑥(𝑡), 𝑦(𝑡)) gives a phase path ordered in the 𝑥𝑂𝑦 plane. The equilibrium points are classified according to the behavior of the orbits in its neighborhood. When it is not possible to determine the analytical solution or the equation of the orbits in the phase diagram, the technique of linearization is used. This technique transforms, locally, the properties of a non-linear system into a linear system. As regards the stability of the equilibrium points and the construction of the respective associated phase diagram, they depend on the eigenvalues of the jacobian matrix of the linear system, which can represent nodes, saddle points, centers or spirals.
Description
Keywords
Sistemas dinâmicos Pontos de equilibrio Estabilidade Órbitas Diagramas de fases Dynamical systems Equilibrium points Stability Orbits Phase diagram Matemática, Estatística e Aplicações . Faculdade de Ciências Exatas e da Engenharia