If $\dot r$ depends on $r$ alone and $\dot\theta > 0$, there is nothing to prove. A closed orbit needs $r(t)$ periodic; $r$ is monotone wherever $\dot r \ne 0$; a monotone periodic function is constant. So the closed orbits are exactly the circles $r = $ root of $f$ in $r > 0$, and the count is a root count. Dulac is not needed and index theory is not needed.
Applied to Book 6's $\dot r = f(r)(1-e^{-z})$ — the modulation is a positive factor for $z>0$, so the roots are $f$'s own — both canonical closures give exactly one, at $r=1$. That is the whole answer for that model, and it took a root scan.
Volume II's toy model is $\dot r = r(1-r^2) + 2(r-1)e^{-z}$. Write $a = 2e^{-z}$ and factorise — block [5] checks this is an identity:
Besides $r=1$ there is a second positive root. The frozen slice has two circular orbits, not one:
and the two radial eigenvalues are $a-2$ at $r=1$ and $(1-r_2)\sqrt{1+4a}$ at $r=r_2$. Both checked against numerical differentiation at six values of $z$.
| z | a = 2e−z | r₂ | eigenvalue at Γ | eigenvalue at r₂ |
|---|---|---|---|---|
| 1.50 | 0.446260 | 0.33442215 | −1.553740 | +1.110746 |
| 0.75 | 0.944733 | 0.59303847 | −1.055267 | +0.889649 |
| 0.25 | 1.557602 | 0.84447074 | −0.442398 | +0.418209 |
| 0.00 | 2.000000 | 1.00000000 | 0.000000 | 0.000000 |
| −0.25 | 2.568051 | 1.17870511 | +0.568051 | −0.599986 |
| −0.75 | 4.234000 | 1.61754576 | +2.234000 | −2.615363 |
Book 6's model has no second orbit at all. The two toy models the series runs in parallel are not variants of one another: a multiplicative modulation $f(r)(1-e^{-z})$ preserves the root set, an additive coupling $+\,a(r-1)$ adds a root. Whichever is meant, the difference has to be stated where both are cited.
The moment $\dot r$ depends on $\theta$, the free count is gone. Strogatz's Example 7.3.1 is the case at hand, because it perturbs the corpus's own field:
Plain Bendixson fails: $\nabla\cdot F = 2 - 4r^2 + 2\mu\cos\theta$ changes sign at $r = 1/\sqrt2$. A Dulac function is needed, and Strogatz is candid on p. 204 that there is no algorithm for finding one, listing $g = 1$, $1/x^ay^b$, $e^{ax}$, $e^{ay}$ as the candidates that occasionally work. The polar analogue of the second works here:
strictly negative on all of $r>0$ exactly when $|\mu| < 1$ — the same threshold under which Example 7.3.1's trapping annulus works. Block [2] checks the closed form against numerical differentiation at nine points, scans a grid at six values of $\mu$, and confirms the sign flips once $\mu = 1.5$.
Dulac as printed requires a simply connected region and concludes no closed orbits. The punctured plane is not simply connected, and there is a closed orbit, so the theorem does not apply as written. Two things bridge the gap.
First, Theorem 6.8.2, p. 180: any closed orbit must enclose fixed points whose indices sum to $+1$. Block [3] scans $r \in (0.01, 4]$ on an $800\times720$ grid and finds $\min|F| = 0.0168$, so the origin is the only fixed point; the winding number on circles of radius $0.05$ through $5.0$ is $+1$ on every one. A closed orbit avoiding the origin would enclose nothing and sum to zero. So every closed orbit encircles the origin, and any two are nested with an annulus between them inside $r > 0$.
This is not the theorem printed on p. 204. It is that theorem's proof run on an annulus instead of a disc, and it is the form that gives uniqueness rather than absence. A reader should check the line rather than take it on the citation.
Existence for $|\mu| < 1$ is Example 7.3.1's trapping annulus, checked in WP-122 block [7]. Uniqueness is the paragraph above. Together, for every $|\mu| < 1$, the perturbed system has exactly one closed orbit. Block [4] corroborates by integrating from five starting radii between $0.05$ and $3.0$ at three values of $\mu$ and finding a spread of $10^{-13}$ on the return ray — corroboration, not proof: three values of $\mu$ against a theorem that covers all of them.
The argument above was built here from index theory and an annulus form of Dulac. There is a classical theorem that reaches the same kind of conclusion without either, and it was published in 1928: Liénard's Theorem (Strogatz p. 212). For $\ddot x + f(x)\dot x + g(x) = 0$ with $f$ even, $g$ odd and positive on $x>0$, and $F(x) = \int_0^x f$ having exactly one positive zero at $x = a$, negative below it and positive and nondecreasing above it with $F \to \infty$, the system has a unique, stable limit cycle surrounding the origin. Strogatz cites Jordan & Smith (1987), Grimshaw (1990) and Perko (1991) for the proof.
The two routes are not interchangeable, and the difference is recorded in ch-van-der-pol. Dulac with $g = 1/r^3$ works here because this cycle is a circle and the field is radial plus rotation; on van der Pol, whose cycle is not a circle, the same $g$ changes sign on the same grid and the route fails. Liénard's theorem never assumed a circle. This paper's result stands as proved; what it does not have is reach.
Book 6's model has $\dot z = 1$ identically, so $z(t) = z_0 + t$ is strictly increasing and no trajectory returns to its starting point. That flow has no periodic orbit at all. $\Gamma = \{r=1\}$ is a helix, and a helix is a closed orbit of the $(r,\theta)$ projection, not of the flow — which is the same fact WP-122 reports as the absence of a return map, seen from the other side.
Vol II's $\dot z = r^2 - 2(r-1)^2e^{-z}$ is $1$ on $\Gamma$ and positive on a tube around it: block [7] scans $|r-1| \le 0.4$, $z \in [0,12]$ and finds $\min \dot z = 0.040000$, at $(r,z) = (0.600, 0.00)$, with the analytic bound $(1-\delta)^2 > 2\delta^2$ holding whenever $\delta < 1/(1+\sqrt2) = 0.4142$. So no periodic orbit lies in that tube either. Outside it $\dot z$ does change sign for Vol II, so this is a statement about a neighbourhood of $\Gamma$; Book 6's $\dot z = 1$ needs no neighbourhood.
Carried into the source, 2026-09-15. WP-22's abstract and §2 now state that the flow has no periodic orbit and that $T^*$ is the period of the $(r,\theta)$ projection; the revision also names the sense of “degenerate,” says where $z$ is frozen in Theorem 5.1, and cites Example 7.1.1 where its data are used. Listed in that paper's own Corrections section, dated. No result was withdrawn.