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A <critical point of a dynamical system> or <equilibrium> is a point $(a,b)$ with $f(a,b)=g(a,b)=0$, so the constant function $(x,y)=(a,b)$ is a solution. Put $u=x-a$, $v=y-b$. For a continuously differentiable <vector field>, its <linearization> is
$$
\frac{d}{dt}\begin{pmatrix}u\\v\end{pmatrix}=M\begin{pmatrix}u\\v\end{pmatrix},\qquad M=D(f,g)(a,b),
$$
with remainder $o(\sqrt{u^2+v^2})$ in the original nonlinear system. For distinct <eigenvalues>, $M$ is diagonalizable over $\mathbb C$, and the linear solutions are combinations of $e^{\lambda_jt}w_j$, where $w_j$ are corresponding <eigenvectors>. Real parts determine growth or decay; nonzero imaginary parts give rotation. For <hyperbolic equilibria>, meaning no <eigenvalue> has zero real part, the <Hartman-Grobman theorem> transfers the local stability and qualitative orbit pattern to the nonlinear system. A purely imaginary pair is not hyperbolic: it determines a linear centre, while higher-order terms decide whether the nonlinear system actually has a centre.
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