= Linearization of a dynamical system
{title2=$\dot\eta=Df(x_*)\eta$}
{wiki}
For a <differential equation> $\dot x=f(x)$ with an <equilibrium point> $x_*$ and differentiable $f$, write $x=x_*+\eta$. The first-order <Taylor expansion> gives
$$
\dot\eta=Df(x_*)\eta+o(\|\eta\|).
$$
The <linearization> discards the smaller remainder and uses the <Jacobian matrix> $Df(x_*)$ to study small perturbations. Its <eigenvalues> predict exponential decay or growth when their real parts are nonzero.
For example, $\dot x=-x+x^3$ linearizes at zero to $\dot\eta=-\eta$, predicting local attraction. A zero <eigenvalue> makes the linear test inconclusive: $\dot x=-x^3$ and $\dot x=x^3$ have the same zero linearization but opposite stability behavior. Thus a <linearization of a dynamical system> supplies a local approximation with hypotheses, rather than a global replacement for the nonlinear dynamics.
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