The Map on Page Ten

Open Nonlinear Dynamics and Chaos at printed page 10. There are no equations on it. There is one figure, and the figure is the book.

Along the top, the number of variables you need to fix the state of a system: n = 1, n = 2, n ≥ 3, n >> 1, Continuum. Down the side, two rows: Linear and Nonlinear. Ten cells. Into them Strogatz drops sixty-one named systems. The top-left corner holds the RC circuit. The bottom-right holds Life.

Across that bottom-right region he writes The frontier, and on the next page he explains the joke. “It’s like in those old maps of the world, where the mapmakers wrote, ‘Here be dragons’ on the unexplored parts of the globe.”

That is the method of the figure. It is a map drawn so the blank parts stay visible as blank. A table of contents cannot do that, because a table of contents has no empty cells.

Neither axis is a difficulty scale. The horizontal one is the dimension of the phase space, and Strogatz says so outright: “Equivalently, this number is the dimension of the phase space.” The vertical one is whether superposition holds.

He teaches both axes with four systems and no algebra. Exponential growth sits at n = 1, because one number predicts the population forever. The pendulum needs two, because knowing the angle does not tell you which way it is swinging. Then the pair that does the real work: “an RC circuit has n = 1 and cannot oscillate, whereas an RLC circuit has n = 2 and can oscillate.” Two components, two sentences, and the axis means something.

This corpus has thirteen volumes and several hundred chapters, and it had never once been placed on Figure 1.3.1. That is strange. It is a nonlinear-dynamics corpus, the figure is the standard chart of nonlinear dynamics, and ch-strogatz already shows that the attractor underneath the whole framework is Strogatz's Example 7.1.1 with a third coordinate added.

So: sixty-one entries, sixty-one patterns, counted with tools/corpus_count.py. That tool reads Poincaré as Poincaré and refuses to read bottom as Bott.

Forty-four entries occupied, seventeen empty. That is better coverage than the corpus had earned. The shape of it is the part worth looking at.

Eight of the seventeen gaps fall in the linear row, which has only eighteen entries in it. They are not exotic topics. They are the RC circuit, the RLC circuit, mass and spring, the two-body problem, coupled harmonic oscillators, equilibrium statistical mechanics, radioactive decay and viscous fluids. A corpus that computes a Conley index and cannot point at an RC circuit has skipped a floor. That floor is WP-82's subject. Figure 1.3.1 finds it in one pass, because Strogatz put those systems on the page to give the rest of it a scale.

Eight of the sixty-one entries have a cell Strogatz states in words rather than leaving to the layout. They are the only eight this page can honestly assign. Here they are with what the corpus holds.

Look at the two zeros. They are the RC and RLC circuits, which is to say they are the two systems Strogatz uses to explain what the horizontal axis means. Of the eight entries whose cell its author bothered to state in prose, the only two this corpus cannot say a word about are the pair that make the axis mean anything.

The other six read 260, 119, 71, 69, 6 and 3. The two largest both sit at n = 1 in the nonlinear row. That is the corner page 11 tells you to start from: “in this book we start in the lower left corner and systematically head to the right.” Nobody told this corpus to obey that sentence. Its density obeys it anyway.

Then Plasmas came back at 93, and the reading stopped.

Ninety-three would make plasma physics the corpus's fourth-largest subject, ahead of chaos. It is not. Open the files and you find plasma cells in the immunology chapters, blood plasma, plasma membranes, and a navigation link that reads Plasma. Some hits are real: stellar plasma and plasma drag, in the Enceladus work. The pattern was spelled correctly, anchored correctly, and entity-aware. It was counting a different subject.

Three ways of miscounting this corpus were already on record in WP-82 §4. A wrong spelling, so k-theory misses $K$-theory. Substring inflation, so gns returns 68 files on designs and assignments. HTML entities, so Liénard matches neither spelling of the name. Each one hands you a number you can see is wrong.

Seven entries were then put through a sense test. A file counts toward the figure's entry only if it also carries a term that fixes the dynamical meaning, and against it if it carries one that fixes another meaning. This is weaker than reading every page. It is reproducible, and both numbers get printed.

Life is the best of these. Strogatz put it in the bottom-right corner, inside The frontier, as the hardest named thing on the page. The corpus returns 151 files. In 140 of them the word is just the English word, in an ordinary sentence. Eleven files are about the origin of life, which is the entry he meant. Eleven is the honest number.

One rule came out of building that test, and it is the part worth keeping. Print the breakdown, not the total. A companion pattern reports a single number, and a single number cannot show you that one loose term inside it is matching three hundred times on something else. The per-term counts show it in one line.

The gaps are the reason to do any of this, and two shapes keep recurring in them.

Some entries are occupied by person and empty by phrase. The corpus has written Smale, and Levinson, and van der Pol. Smale appears in 12 files and Levinson in 6. It cannot say forced nonlinear oscillator, which reads 0, or anharmonic, which also reads 0. It knows the people and not the vocabulary. That is the inverse of the usual failure, and it is what a corpus built by reading biographies looks like from the outside.

The rest of the pattern is the linear row. Its gaps are the first systems in an undergraduate course. Radioactive decay is the simplest entry in the whole figure and it reads 0. Spatio-temporal complexity reads 0 while reaction-diffusion, which lives in the same region, reads 26. Practical uses of chaos reads 0 while chaos reads 69, so the applied entry of the corpus's largest subject is the empty one. Josephson appears in 4 files and Josephson arrays in none. Lasers appear in 11 and nonlinear optics in none.

Neither pattern was visible from inside the corpus. Both took one afternoon with somebody else's figure.

One thing has to be said about method, because it decides which numbers this page is allowed to print.

The figure is set rotated on the page, so every glyph carries a position and the grid can be recovered rather than guessed. Two hundred and thirty-six positioned text runs make up page ten. Five column labels sit on a single line at x = 163.2, and the fifth is Continuum. The two row labels sit at x = 217 and x = 372. Linear-row glyphs occupy one band and nonlinear-row glyphs another, and the bands do not overlap: exactly one of the 236 runs falls between them. So the row of every entry is recoverable.

The column is not. The extractor hands back one text run for a whole row-line spanning several columns, so 'Fixed points Pendulum Strange attra' arrives as a single string at a single coordinate. Fifty-three of the sixty-one entries therefore have no column this page is entitled to publish, and none is published. A reader holding the printed book sees every cell at once. Position is what a figure is for, and the strings are only captions.

Six italicised phrases do survive, the ones marking where a class of behaviour first arises. Each is a short run with its own coordinate, and a label sits about 23 units before the column it heads. Calibrating that offset on the four labels Strogatz confirms in prose puts Collective phenomena over n >> 1 and Spatio-temporal complexity over Continuum. The frontier lands between two markers and is left unassigned. Three cells carry no class label at all, which is what you would expect if the labels mark first arrivals.

Strogatz drew the figure so that the empty parts would stay visible. He was pointing at the frontier of the subject, and he put dragons on it.

Laid over this corpus, the same page points somewhere less romantic. The blanks are not at the frontier. They are in the top-left corner, among the circuits and springs that every course starts with, and they sit next to entries reading 260 and 121. The corpus went to the hard cells first and left the scale behind it unbuilt.

That is a useful thing for a map to be able to say, and no page inside the corpus could have said it. It took somebody else's figure, drawn thirty-two years ago for a different purpose, and an afternoon.

Strogatz	S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview 2015, CRC/Chapman & Hall 2018 printing. Figure 1.3.1 and caption, printed p. 10. The framing prose is §1.3, pp. 9 and 11: the axes, the exponential-growth and pendulum placements, the RC/RLC contrast, the reading order, and The frontier with the here-be-dragons remark. First edition 1994.
instrument	tools/corpus_count.py, entity-aware and anchored, tracked files only. Its docstring carries the first three failure modes; this page added the fourth.
in-corpus	ch-strogatz · ch-smale · ch-van-der-pol · ch-euler · WP-82 · The Missing Floor
verification	book7/ch-the-map-on-page-ten-verify.py, nine blocks, standard library only. The PDF blocks skip cleanly without the book. Every number on this page is printed by it.