Near a fixed point a smooth nonlinear system looks linear, and often that is all you need. Sometimes it is exactly what you must not trust. Four cases, each with the numbers, and a plate that overlays the true system on its linearization.
The object. A nonlinear system in the plane (white), its linearization at a fixed point (dashed pink), and the point you start from. Drag the start and watch where the two stay together and where they part.
Zooming in on a smooth curve makes it look like a line. The question is what the line is allowed to tell you. It is allowed to tell you the truth about a small neighbourhood, for the sort of thing the neighbourhood can decide, and nothing about what happens outside it, or about questions the linear part is too coarse to settle.
Before trusting the linearization of x′ = f(x) at a fixed point x*, check:
(a) x* really is a fixed point (or the linearization is about a known solution). (b) No eigenvalue of the Jacobian Df(x*) has real part zero (the point is hyperbolic). (c) The neglected terms stay small over the amplitudes and times you care about, which you can test by simulating the full system. If (b) fails, the linearization is silent. If (c) fails, it is simply wrong there.
(b) is the hypothesis of the Hartman–Grobman theorem, stated and discussed with its page references in Book XXI Chapter 3 (Strogatz, printed pp. 154–156; Perko §2.8, pp. 119–120). That chapter also verifies the numbers used below; this interlude collects them as a decision guide. The theorem gives a topological statement (the kind of fixed point), not rates.
Select the damped pendulum and start near the origin. The white and pink curves lie on one another. Drag the start out and they separate; drag it past the dotted line at θ = π and the true system no longer goes to the origin at all.
At the origin the Jacobian has eigenvalues −0.1 ± 0.995i, off the imaginary axis, so (b) holds, and Hartman–Grobman says what the plate shows near the origin. It says nothing about the basin: the pendulum’s other fixed points (saddles at ±π, more sinks at ±2π) are invisible to a linearization at 0. A local theorem cannot answer a global question.
Pick Example 6.3.2. The linearization at the origin is a pure rotation: circles forever. The true system spirals in, so slowly that over short records it passes for a rotation, or a decay at an ordinary rate.
Let r² = x² + y². Then r r′ = x x′ + y y′ = x(−y + a x r²) + y(x + a y r²) = a r&sup4;, so r′ = a r³. And r²θ′ = x y′ − y x′ = x(x + a y r²) − y(−y + a x r²) = r², so θ′ = 1. Separating variables in r′ = a r³: d(r−2)/dt = −2a, so r−2 = r0−2 − 2at, which gives both formulas. □
Computed: with a = −1 and r0 = 1, r = 0.2182 at t = 10, 0.0705 at t = 100 and 0.0224 at t = 1000, against r = 1 from the linearization. Reaching r = 0.001 takes 499,999.5 time units, against 15.7 for a hyperbolic sink with rate 0.44, a ratio above 30,000. For a = +1 the same linearization predicts the same circles while the true solution leaves in finite time. One linearization, two opposite truths: the nonlinear terms decide.
As a stable equilibrium is pushed toward a fold, the linear restoring force weakens until it vanishes. The theory built on eigenvalues is working right up to the point and then has nothing to say at it.
The fold is a marginal case, Strogatz’s second (a zero eigenvalue). Hartman–Grobman covers every point on the approach and is silent at the point. The tools for that point are normal forms and bifurcation theory (Strogatz chapters 3 and 8). In this series the fold file Orthogenesis/Taxonomy/FoldHysteresis.lean records, for the cusp slice Wa(q) = q⁴/4 − 3q²/2 + aq, that W″ = 3q² − 3 vanishes at q = 1 and that a stable equilibrium with q > 1 exists exactly when a < 2; the author’s kernel report for it is dated 25 Sep 2026, and it was not re-run for this chapter. Book XXI Chapter 3 adds that the series’ own fold operator is piecewise linear with a corner, so it has no linearization there at all.
In a chaotic system the linearization along a trajectory tells you how fast nearby trajectories separate, but only while they are still near. Once the separation is large the linear picture has no say, and the folding that keeps the motion bounded, which is nonlinear, takes over.
Δ is a choice, not a law of nature: it is the scale at which you decide nonlinearity can no longer be ignored. A correction recorded in this corpus (Orthogenesis/Lyapunov/LyapunovHorizon.lean, Book 3 week 10) shows how rates get mixed up: λ = 0.9 per day grows by e9 ≈ 8103 over ten days, while “doubles every day” is λ = ln 2 and grows by 210 = 1024, about eight times less. The Lean file’s header says untested; an axioms report from 25 Sep 2026 exists, and I have not re-run it.