Every model in this corpus is studied through its linearisation, the Jacobian at a fixed point. The Hartman–Grobman theorem says when that is legitimate: near a fixed point with no eigenvalue on the imaginary axis, the linear picture is the true picture up to bending. The fine print is where the corpus lives. The theorem is silent exactly at thresholds, folds and bifurcations, which are the points this series is about.
Strogatz states it on p. 156. A fixed point is hyperbolic if all eigenvalues of the linearisation have nonzero real part. The theorem says that “the local phase portrait near a hyperbolic fixed point is ‘topologically equivalent’ to the phase portrait of the linearization; in particular, the stability type of the fixed point is faithfully captured by the linearization” CITED. Chapter 2 showed how coarse topological equivalence is. So even at its strongest the theorem licenses a qualitative statement: sink, source or saddle. It does not license the rates.
Perko states the theorem with its hypotheses in full (§2.8, pp. 119–120). The vector field must be C¹, the fixed point is moved to the origin, and Df(0) must have “no eigenvalue with zero real part.” Then there is a homeomorphism between neighbourhoods of the origin that carries the nonlinear flow onto the linear one and preserves the time parameter, a local conjugacy in Chapter 2's sense. The C¹ hypothesis matters for §4: a vector field with a corner is not covered at all CITED.
A page earlier (p. 155) Strogatz sorts fixed points into robust cases (sources, sinks, saddles) and marginal cases: centers, and points where “at least one eigenvalue is zero”. Only the robust cases are covered. Centers, he writes, “live on the razor's edge between stability and instability.”
His Example 6.3.2 (p. 154) is built to break it: x′ = −y + a x(x²+y²), y′ = x + a y(x²+y²). The linear part is a pure rotation, so the linearisation predicts a center: closed orbits, r constant forever. In polar coordinates the system is r′ = a r³, θ′ = 1. For a < 0 it spirals in. The script integrates it with a = −1 from r = 1:
| time | r, integrated | r, closed form 1/√(1+2t) | linearisation's r |
|---|---|---|---|
| 10 | 0.218218 | 0.218218 | 1 |
| 100 | 0.070535 | 0.070535 | 1 |
| 1,000 | 0.022355 | 0.022355 | 1 |
The linearisation gets the kind of fixed point wrong COMPUTED. And the true system spirals in the way Strogatz warns it will, “extremely slow”: algebraically, not exponentially.
Set Example 6.3.2 beside a hyperbolic sink with the corpus's own immune rate, μ = −0.44:
| hyperbolic, μ = −0.44 | marginal, Example 6.3.2 | |
|---|---|---|
| time to halve r | 1.58 | 1.50 |
| time to reach r = 0.001 | 15.7 | 500,000 |
Early on they look alike. Halving takes about the same time, which is exactly how a marginal system passes for a robust one on a short record. Near the fixed point they part by a factor of over 30,000 COMPUTED. A model fitted on the early data, reporting a decay rate, would be describing a rate that does not exist.
This series is organised around thresholds: the fold, κ*, the transition. At a fold, the textbook's saddle-node x′ = r + x², the stable fixed point sits at x* = −√(−r) with eigenvalue 2x*. The script walks r toward zero: the eigenvalue goes −2, −0.2, −0.02, and at the fold it is exactly zero COMPUTED. That is Strogatz's second marginal case. Hartman–Grobman covers every point on the approach and says nothing at the point itself.
The corpus's own fold operator, F₃ in AMonster/dm3_operators.lean, is worse placed still. As Book XVIII ch. 2 found, it is piecewise linear with a corner at r = κ. It has no linearisation there at all, hyperbolic or not. The places where this series makes its claims are exactly the places where the standard licence for linear reasoning does not apply. That is not a reason to stop. It is a reason to use the tools built for those points — normal forms and bifurcation theory, Strogatz chapters 3 and 8 — rather than the Jacobian. OPEN — the corpus's fold claims have not been re-derived with those tools.