By examining the signs of the eigenvalues of the linearization of the equilibria's equations, equilibria can be categorized. In other words, the equilibria may be classified by evaluating the Jacobian matrix at each of the system's equilibrium points and then determining the resulting eigenvalues.
Then, by locating the eigenvector(s) associated with each eigenvalue, the behaviour of the system in the vicinity of each equilibrium point can be qualitatively (or even statistically, in some cases) identified. If none of the eigenvalues at an equilibrium point have zero real component, the equilibrium is hyperbolic.
The point is stable if all of the eigenvalues have negative real portions. The point is unstable if at least one has a positive real part.
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