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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 iswith 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 $e^{{\lambda }_{j}t}{w}_j$, 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.