Retratos de fase de sistemas de equações diferenciais lineares com coeficientes constantes de até terceira ordem

Abstract

This study was developed with the intention of investigating the phase portraits of all linear systems with constant coefficients up to third order. For this purpose, we will determine the systems solutions, perform an analytical study of them and sketch illustrations representing the portraits. Then, we may identify similarities and differences and classify the phase portraits according to the topological behavior, understanding the importance of this kind of classification. To achieve these goals, we will go through preliminaries that permeate first order linear differential equation, methods for finding solutions, linear systems, eigenvalues and eigenvectors, exponential matrices and Jordan matrices. Finally, we conclude that there is a large plurality of phase portraits, which increase with the dimension. For first dimension, we have three phase portraits that are geometrically different from each other, while for the second dimension, we have thirteen phase portraits and, for the third dimension, thirty seven portraits. Therefore, we understand the importance and facilitation of these topological conjugations, reducing the number of cases to three, seven and thirteen, for first, second and third orders, respectively.