# Hartman-Grobman Theorem

Suppose $(\bar x,\bar y)$ is a fixed point of a system

$$\begin{cases} \dot{x} = f(x,y)\\ \dot{y}= g(x,y). \end{cases}$$

Assume that the real part of the eigenvalues of the Jacobian matrix

$$A= \begin{pmatrix} \frac{\partial f}{\partial x}\scriptstyle{(\bar x,\bar y)} & \frac{\partial f}{\partial y}\scriptstyle{(\bar x,\bar y)}\\ \frac{\partial g}{\partial x}\scriptstyle{(\bar x,\bar y)} & \frac{\partial g}{\partial x}\scriptstyle{(\bar x,\bar y)} \end{pmatrix}$$

are nonzero. Then there is a small region around $(\bar x,\bar y)$ on which the phase portrait for the original system is topologically equivalent to the phase portrait of the linearized system in a small region around $(0,0)$.