linearization about an equilibrium

Suppose a system is sitting almost still at a resting state — a pendulum hanging straight down, a population parked at its carrying capacity. If you nudge it just a little, the messy nonlinear forces that govern it are, to a first approximation, well described by a much simpler linear system. Linearization is the act of throwing away everything except that first approximation, so you can use the full machinery of linear theory to predict what happens right near the resting state.

Here is the recipe in plain steps. Start with an autonomous system x' = f(x), and let x* be an equilibrium, meaning f(x*) = 0 so the system would stay there forever. Write the state as x = x* + u, where u is a small deviation. Then a Taylor expansion gives f(x* + u) approximately equal to f(x*) + J u, where J is the Jacobian matrix of f evaluated at x* — the table of all the first partial derivatives. Since f(x*) = 0, the deviation obeys u' = J u, a plain constant-coefficient linear system. You analyze it with eigenvalues, exactly as in the linear phase plane.

This matters because most real systems are nonlinear and have no formula for their solution, yet near each equilibrium they behave like a linear system you can fully solve. The honest catch: linearization only tells you about behaviour very close to the equilibrium and only when that equilibrium is hyperbolic — when no eigenvalue of J sits exactly on the borderline (zero real part). At a borderline case the discarded nonlinear terms can flip the answer, so the linear picture is then untrustworthy.