An Undergraduate’s Guide to the Hartman-Grobman and Poincaré-Bendixon Theorems
نویسنده
چکیده
The Hartman-Grobman and Poincaré-Bendixon Theorems are two of the most powerful tools used in dynamical systems. The Hartman-Grobman theorem allows us to represent the local phase portrait about certain types of equilibria in a nonlinear system by a similar, simpler linear system that we can find by computing the system’s Jacobian matrix at the equilibrium point. The Poincaré-Bendixon theorem gives us a way to find periodic solutions on 2D surfaces. One way in which we can use this theorem is by finding an annulus-shaped region (2D donut shape) such that the vectors on both edges point into the region. This document is a guide to the proofs of these two powerful theorems. These proofs are not generally covered in dynamical systems courses at the undergraduate level. Many such courses do not require previous knowledge of topics such as mathematical analysis and topology. This guide is intended to be a self-contained explanation of the proofs of these theorems in the sense that it should be comprehensible to those who have a basic understanding of set theory, calculus, linear algebra and differential equations and who are currently studying dynamical systems.
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