the Hartman-Grobman theorem
/ HART-man GROB-man /
Linearization gives you a simple linear system that approximates a nonlinear one near an equilibrium. But is the linear picture actually right, or just a hopeful sketch? The Hartman-Grobman theorem is the guarantee. It says that, away from borderline cases, the true nonlinear flow near the equilibrium is a gently bent copy of its linearization — same qualitative behaviour, just drawn on warped paper.
Precisely: if x* is a hyperbolic equilibrium — every eigenvalue of the Jacobian J there has nonzero real part — then there is a continuous one-to-one change of coordinates, defined in a small neighbourhood, that turns the nonlinear trajectories into the trajectories of the linear system u' = J u. 'Topologically conjugate' is the technical phrase: you can deform one phase portrait into the other by a continuous stretch without tearing. So a saddle stays a saddle, a stable node stays a stable node, a stable spiral stays a stable spiral — the count of incoming and outgoing directions, and whether things spiral, are all preserved.
This is why linearization is trusted in practice: at a hyperbolic equilibrium, computing the eigenvalues of one matrix settles the local question completely, no nonlinear analysis needed. The theorem is also honest about its boundary. It says nothing when the equilibrium is non-hyperbolic (some eigenvalue has zero real part). There a center of the linearization might be a spiral of the true system, or a borderline could hide a slow drift — the nonlinear terms you threw away now decide the verdict, and you must look closer.
At a saddle of x' = x + y^2, y' = -y, the Jacobian [1, 0; 0, -1] has eigenvalues 1 and -1, both real and nonzero, so the equilibrium is hyperbolic; Hartman-Grobman guarantees the curved nonlinear trajectories form a genuine saddle.
Nonzero-real-part eigenvalues make the point hyperbolic, so the nonlinear flow is a bent copy of its linear model.
The theorem only promises a continuous (not necessarily smooth) change of coordinates, and it says nothing at non-hyperbolic equilibria — the classic trap is a linear center, where the nonlinear truth could be a stable spiral, an unstable spiral, or a genuine center.