*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 $(0,0)$.