Strogatz prints two examples of a limit cycle one page apart. The first is this corpus's transverse attractor. The second is van der Pol's, and the difference between them is not decoration — one cycle is a circle and the other is not, which is precisely what the closure conditions require and therefore rule out.
ch-strogatz establishes that the corpus's transverse system is Example 7.1.1, p. 199. One page later, on p. 200, Strogatz gives the second example of the same section:
Counted on tracked *.html and *.md at HEAD, and counted in
chapters rather than files — because a file count also counts index pages,
the audit log and this chapter, which is
WP-82 block [3]'s finding and it
applies here too, measured the day this chapter was written:
“limit cycle” appeared in 108 chapters, “van der Pol” in
none, “Liénard” in none, “memristor” in none. The
raw file counts were 119 and 5 and 2 and 3; the difference between the two columns is
entirely listings and this page. The phenomenon is everywhere and
the second canonical instance of it is absent. That is not a gap in citation — the
corpus derives rather than reads — but it is a gap in the worked examples, and the
next two sections are what it costs.
Two of those zeros are no longer zero, which is the outcome block [6] was built to detect. “van der Pol” is now in two other chapters — WP-120, which cites Liénard's Theorem for the reach the Dulac route lacks, and ch-conley. The assertion was updated to name that exact set rather than relaxed, so a third arrival still fails and still notifies.
The Liénard row was a false zero. This page
writes the accented e of the name as an HTML character entity rather than as a
literal, and so does WP-120, so the pattern li[eé]nard matched neither
of them in source — a zero reported for a word printed
twice on the page doing the counting. Sixty-eight tracked HTML files carry accented
entities; entity-aware, Poincaré reads 68 rather than 58, Gödel 27 rather than
23, and Liénard 5 rather than 2. The instrument is
tools/corpus_count.py; block [6] now uses its pattern.
Block [2] settles the orbit and then measures it over one full turn:
| μ | rmin | rmax | rmax/rmin | period |
|---|---|---|---|---|
| 0.1 | 1.9369 | 2.0668 | 1.067 | 6.2870 |
| 1.0 | 1.5317 | 2.8300 | 1.848 | 6.6630 |
| 1.5 | 1.4134 | 3.3608 | 2.378 | 7.0960 |
| 5.0 | 1.1825 | 7.7015 | 6.513 | 11.6120 |
Example 7.1.1, for contrast, needs no integration at all: $\dot r = r(1-r^2)$ has $r=1$ as a fixed point and $\dot\theta = 1$ advances the angle at unit rate, so the orbit is the unit circle and the period is $2\pi$ — ratio $1.000$, period $6.283185$, both exact. The distortion in the table grows monotonically with $\mu$, and by $\mu = 5$ the waveform is a relaxation oscillation whose period (Strogatz §7.5, p. 213) grows linearly in $\mu$ rather than staying put.
This is not a defect. It is a restriction the family imposes and has not named: it is circle-preserving, and everything WP-22 proves is proved about circular cycles. Naming it costs one clause and tells a reader exactly how far the results reach.
WP-120 proves uniqueness with index theory plus Dulac's criterion, using $g = 1/r^3$. Run the same $g$ on van der Pol, on the same grid — block [5]:
The corpus's route works because that cycle is a circle and the field is radial plus rotation. It does not survive the move to a non-circular orbit. What does is Liénard's Theorem, Strogatz p. 213 — five conditions on $f$ and $g$ in $\ddot x + f(x)\dot x + g(x) = 0$, and then a unique, stable limit cycle surrounding the origin. Block [1] checks all five for van der Pol: $g(x)=x$ is odd and positive on $x>0$, $f(x) = \mu(x^2-1)$ is even, and
has its unique positive zero at $a = \sqrt3$, is negative on $0<x<a$, and is positive and nondecreasing beyond it. That is Example 7.4.1, p. 213, and the theorem it invokes is Liénard 1928. Two correct routes to the same kind of conclusion; only one of them was built without assuming the answer's shape.
Strogatz opens §7.4 by saying where Liénard's equation came from:
In the early days of nonlinear dynamics, say from about 1920 to 1950, there was a great deal of research on nonlinear oscillations. The work was initially motivated by the development of radio and vacuum tube technology, and later it took on a mathematical life of its own.
— Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., p. 212van der Pol's equation is a vacuum-tube circuit. Figure 1.3.1, p. 10, puts it in the nonlinear $n=2$ cell under the heading “Nonlinear electronics (van der Pol, Josephson)”, and the nonlinear $n\ge3$ cell — the chaos cell — is where Chua's circuit lives. Nonlinear electronics is the field that produced the mathematics this corpus runs on, and the corpus contains almost none of its names: van der Pol 0, Liénard 0, memristor 0, and “Chua” in two files, both of them the game-theory pack. Those are chapter counts; block [6] prints the composition beside them.
That is the method WP-82 runs on its rung ladder and that Figure 1.3.1 runs on its cells: enumerate the relations, find the empty one, treat the gap as a prediction. Chua's is the strongest instance of it known — gap found 1971, filled 2008. It is cited here as a precedent for the method, not as a result about this corpus.
[OPEN] pending measurement on a physical basin. Whether a
memristor's loop is that fold or a different mechanism is an equation-level question,
not a resemblance, and it is recorded as row N11 of
the novelty register in the state it is
actually in: unsearched.
The register for a van der Pol citation here is corrective and generative at once. Corrective, because the closure family turns out to be narrower than it reads and the uniqueness argument narrower still. Generative, because the obvious next question is now well posed: what does the contact construction look like over a non-circular cycle? Drop $\dot\theta = 1$, or let the radial speed depend on $\theta$, and every quantity the series computes — the period, the multiplier, the neutral line — has to be recomputed rather than read off. Liénard's theorem says the cycle will still be there and still be unique. Nothing in the corpus says what happens to the contact form.
Strogatz supplied the object the corpus is a modification of. van der Pol supplies the nearest object it is not, and that is the more useful of the two, because it marks an edge rather than a floor.
| Strogatz 2015 | S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview 2015 / CRC 2018, ISBN 978-0-8133-4910-7. Example 7.1.2 p. 200 · §7.4 p. 212 (Liénard's equation, and the section heading) · Liénard's Theorem p. 213 · Example 7.4.1 p. 213 · §7.5 p. 213 (relaxation oscillations) · Figure 1.3.1 p. 10. |
| Liénard 1928 | The uniqueness theorem for $\ddot x + f(x)\dot x + g(x) = 0$. Strogatz cites Jordan & Smith (1987), Grimshaw (1990) and Perko (1991) for the proof. |
| Chua 1971 | L. O. Chua, Memristor — The Missing Circuit Element, IEEE Trans. Circuit Theory CT-18, 507–519. Physical claim: Strukov, Snider, Stewart & Williams, The Missing Memristor Found, Nature 453 (2008). |
| Verification | book7/ch-van-der-pol-verify.py — 7 blocks, standard library only, exit 0. Liénard's five conditions with $a=\sqrt3$; the orbit measured at four $\mu$; Example 7.1.1's exact circle and exact period; the closure-conditions argument; the Dulac sign test on a 5,400-point grid; the corpus counted in chapters rather than files, with the composition of every hit printed; a control block. |
| Internal | ch-strogatz · WP-120 · WP-122 · WP-82 · WP-22 · novelty register |