Summary
Intersectional dynamics describes what happens along a trajectory because of outside forces, patterns and cycles and where it has the potential to go based on what it interacts with. Complex systems exist across orders of magnitude, in time and space. What appears chaotic might be shaped by that interconnectedness along the non-linear path. Much of what is unseen is harder to measure and even if we could see it we don't have the instruments to measure it, but that is all changing. New instruments are expanding what can be known or discovered about what exists, why it exists and even what it's becoming. What we can know often leaves an imprint, and where the imprint can be read, the course can be traced back to where and why it changed, even leaving a basis for anticipating what might happen again. Initial conditions are important, but they're not the whole story and the turn of events might display a chaos that can't be known from the start, including when an intersection or collision resets a trajectory.

Paper 57 — Intersectional Dynamics Theory

Trajectories of Complex Systems

Abstract

A chaotic system that seems disorderly may be the necessary step before creation itself. When systems come into contact with other things, sometimes even only their fields, each meeting has substance and composition, a chemistry, a speed or orbit, an angle and a geometry. The way things travel can be disturbed and line back up, redirect itself or be propelled, travel alongside something else, merge for a time, split and branch off or even go the opposite direction. In a collision it can break apart or transform, and what breaks off goes on to act elsewhere. Randomness can enter without the trajectory collapsing; it keeps going.

1. Chaos

What if chaos had a trajectory that only seemed chaotic while it was getting there? There is a point where something starts out carrying what it is, that pattern, that imprint, for example a human's lifetime of lived experiences in memory, but what if when it intersects with another thing, it receives an imprint it carries forward that is only noticeable by the result or destination? When an impression is made at an intersection what makes it permeable enough to persist? Did the boundary cause something to cohere or travel separately? Did these things meet back up? And can whatever it's carrying be traced back to where it was received?

Chaos theory, in its classic form, defines chaos as a property of deterministic systems. The system follows fixed rules, no randomness enters, and the future is fully set by the starting state. Yet the trajectory can't be followed in practice, because tiny differences in that starting state grow exponentially. Edward Lorenz found this in a weather model in 1963. A summary widely attributed to him puts it this way: the present determines the future, but the approximate present does not approximately determine the future.

That definition holds inside a closed system. Lorenz's model has three variables and three equations, and its trajectory is completely determined by where it begins. Nothing new enters once it starts, and the chaos stays inside a bounded shape. The unpredictability comes only from not being able to measure the starting state precisely enough.

Outside such a model, what arrives has its own cycles: the sun's, the seasons', the tides', Earth's orbit. Models that let outside influences in mostly take them as fixed inputs or noise, without how they cycle. Left out, those cycles look like irregularity, and that irregularity gets called chaos. Included, some of it should turn into timing, a system locking to the cycle that arrives, as El Niño tends to peak at the same time of year. Over the Tiwi Islands off northern Australia, through the months before the monsoon, a deep thunderstorm builds almost every day at around three in the afternoon, regular enough that people there gave it a name, Hector. Sea breezes from either side of the islands meet over the heated land and are forced upward. Weather that should be the least predictable thing in the sky keeps an appointment with the daily heating of the land (Weatherzone). If including the arriving cycles changes nothing, the claim fails for that case.

What looks like chaos in a real system is a course still being shaped by what it meets. Something can be disturbed and line back up. It can be redirected or propelled, travel alongside something else for a while, merge for a time, split and branch off, or turn and go the opposite direction. It can break into a million pieces that each go on to become something. Randomness can enter and the trajectory keeps going, carrying what it went through. Some of this the classic model cannot do at all: in a deterministic system one state leads to one future, so a trajectory there can never split.

Physics has already found irregular motion that the accepted test misses. In what has been named stable chaos, tiny differences die out instead of growing, so by the standard measure there is no chaos, and yet the motion stays irregular, driven by disturbances of real size that work their way in from the system's boundaries (Politi & Torcini, 2009). The simplest picture in that work is a ball bouncing on a floor. Two nearly identical bounces separate for an instant at the impact and close up again afterward. When three particles collide at once, the order of the collisions can flip, and the outcome changes abruptly. With equal masses, nearby paths drift apart slowly; make one mass twice the other and they are far apart within about five collisions. What meets decides how much the meeting matters.

Two kinds of meeting need to be kept apart. A collision is contact. Things hit, and something breaks apart, transforms or breaks off, and what breaks off goes on to act somewhere else. Ice calving from a glacier drifts out and changes the water it melts into; a fragment knocked from an asteroid can travel for millions of years before it burns into another planet's sky. An intersection can happen at a distance, through a field, so it can propel, redirect or disturb without anything touching. A spacecraft swinging past a planet leaves on a new course at a new speed, and the planet's own path shifts by an amount too small to measure. Charged particles streaming from the Sun meet Earth's magnetic field and are turned toward the poles. A magnetometer on the ground swings while they arrive and settles when they pass. The field recovers, and the instrument's record keeps the disturbance.

Every meeting also has a geometry. Waves that reach a beach head-on move sand up and down it; waves that arrive at an angle carry it along the shore, and over years build sandbars and spits or take a stretch of coast away. Light rising through water crosses into the air at shallow angles and, past a critical angle, cannot cross at all and turns back entirely. The same two things meeting at a different angle do something different.

2. Before Chaos Theory

It began as everyday disorder, was attached to a deterministic process in 1975, and was reshaped by each field it entered. Chaos as it occurs in open systems is still being shaped along the way. It can change. The field has a plainer name for itself: nonlinear dynamics. The name describes the mathematics, where a small change need not produce a small effect, and it carries none of the disorder that "chaos" does. Not every nonlinear system is chaotic, but every chaotic one is nonlinear. "Chaos" named one striking behavior inside the field, and the striking word is the one that spread.

Edward Lorenz called it deterministic nonperiodic flow. His 1963 paper, the one that found sensitivity to initial conditions in a model of the weather, carries that title: a flow that follows fixed rules and never repeats. The name came twelve years later, when Tien-Yien Li and James Yorke titled a 1975 paper "Period Three Implies Chaos." A process set entirely by its starting point was given the word for disorder.

The theory has math, and it works; the problem is with it being called chaos. In ordinary language chaos is what comes apart and cannot be pinned down; in the theory it is a course decided at the start, unpredictable only because the start can never be measured exactly. The word carried that deterministic meaning out of mathematics and into ecology, physiology and epidemics, into living systems that stay open and keep being shaped by what they meet. Forty years after the name, a paper marking the twenty-fifth year of the field's own journal opens by saying there is "still no broadly applicable, convenient, generally accepted definition of the term chaos" (Hunt & Ott, 2015).

In 1873, James Clerk Maxwell described singular points: places in a system's course where a cause too small to measure decides which way it goes. His examples included "the little spark which kindles the great forest" and the rock, loosened by frost, balanced on a singular point of the mountain-side. At such a point the system is open to what meets it, including a push that would bring it back on course.

Boole poses the reverse problem in chapter I of An Investigation of the Laws of Thought (1854). Free agency would seem to rule out "orderly evolution" in society, yet the records of crime and pauperism show a regularity, while the seasons, volcanoes, blight and epidemics, things that seem the product of natural causes, "refuse to be submitted to regular and apprehensible laws." "Fickle as the wind," he notes, is a proverb. His answer is that free agency in the individual is not inconsistent with regularity in the system, and that order will not be "perceptible to human observation, unless the play of its producing causes is sufficiently simple."

That is the reverse of chaos theory. In chaos, every part follows fixed rules, and the whole still can't be predicted. In Boole, the parts are free, and the whole is still regular.

Boole's laws of thought led somewhere he did not go. His algebra is one "in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these values alone," and its founding law is x² = x. In 1937 Claude Shannon showed that this algebra describes switching circuits. From thought, a law; from that law, computation.

3. Order

Chaos has magnitudes of order that cycle into each other. This intersection makes or breaks a system depending on that order. It's energy running into geometry.

Mathematicians looking inside chaotic systems found order. A chaotic path never repeats, yet it never leaves its shape, called a strange attractor. That shape is made of repeating cycles set by the system itself; the path follows one for a while, then moves to the next. Those that pass into chaos by doubling their cycles do so at the same ratio, about 4.669 (Feigenbaum, 1978).

Whether that order makes or breaks a system depends on what meets it. The Bay of Fundy's basin has a natural period close to the twice-daily tide. Energy arrives at a frequency the geometry can receive, and the result is the highest tides on Earth, over fifteen meters. In Mexico City in 1985, the old lakebed beneath the city shook at the period of mid-height buildings. Those buildings fell, while shorter and taller ones mostly stood. The geometry was already there. What arrived decided the outcome.

A system can lose access to its own order without the order being gone. A cell carries what it is, and over a lifetime it also accumulates chemical marks on its DNA, which it actively maintains. One possibility, still open, is that the cell ends up remembering the accumulation instead of its own baseline. Reprogramming factors expressed briefly reset some of those age-related marks while the cell stays what it is; expressed longer, they erase its identity. Same factors, different window, opposite outcome.

In 1990, Ott, Grebogi and Yorke showed that a chaotic system can be held on one of its own cycles with small, well-timed pushes: nothing added, only what was already there lined up. That correction comes from outside. A system that carries the pattern of when to correct itself, that recognizes when what has accumulated no longer serves the trajectory, would not need an outside controller. Its consistency would be in when to shed what accumulates. One test: how fast a cell re-ages after reprogramming shows whether the marks were imprint or pattern, kept after the conditions changed or rebuilt because the conditions persist.

4. Signals & Imprints

Imprint and impression come from the same root, the Latin imprimere, to press into. An impression is made at an intersection, where one thing presses into what it meets. The imprint is that impression becoming visible. Many impressions are made and few become imprints. Whether one does depends on whether what received it kept the difference and then stopped changing, as lava does when it cools through the point that fixes the direction of Earth's magnetic field inside it, or a tree does when a hard year closes a narrow ring.

Intersections can be where life forms, and at the genomic level this dynamic system leaves a trace of what intersects. At fertilization the germline marks are erased and reset, yet a small set of genes keep a mark of which parent each copy came from, and that mark steers how the embryo develops. This is genomic imprinting, and it is found in mammals, insects, and flowering plants.

Across the landscape these imprints can be read directly, in rock and crystal and tree rings. In other things the impression stays out of sight, and the instrument that could measure it may not exist yet. It can still be noticed in other ways. It can show in a reaction when the same thing comes back: the immune system carries the impression of an infection it beat, invisible until that infection returns and the response comes faster and stronger. It can show in what it does to something nearby: Neptune was found in 1846 before anyone had seen it, from the way Uranus kept drifting off its predicted path. It can wait and show much later: people who were in the womb during the Dutch famine of 1944 and 1945 showed differences in health decades afterward, and some carry measurable chemical differences on their DNA. It can stay in composition, as the chemistry of tooth enamel keeps a trace of the water and food of the place a person grew up.

Where the impression is a geometry, the way it becomes visible can be followed closely. A single collision can be frozen. An asteroid that strikes the Moon, Mars or Venus at less than about twelve degrees above the surface leaves an elongated crater, and below about ten degrees the material it throws out spreads in two wings along the line it came in on, so the direction of arrival can still be read (Bottke et al., 2000). Repeated contact can be layered. Water and wind moving over sand build ripples with a gentle upstream side and a steep downstream side, and when the sand becomes rock the asymmetry stays, so the direction of a current that ran millions of years ago can be read in a cliff face.

Not every impression becomes readable. Above those angles, craters come out nearly round and the direction of arrival is gone; only four or five percent of craters are elongated. Where different causes leave the same mark, tracing back stops, and that is one of the places this theory can be shown wrong.

In living things an impression also steers what comes next. A young bird does not hatch knowing what it is. It learns its species from what it meets in an early window and identifies with that species for life, which later decides whom it chooses as a mate (Wildlife Center of Virginia). A ewe learns her own lamb by its smell shortly after birth (Burger et al., 2011). Male sheep and goats raised by the other species appear to carry its social and sexual preferences for good, while in females the effect is weaker and can reverse (Kendrick et al., 1998). Here it is more than a record of the last intersection. It becomes a condition of the next one, deciding which intersections are recognized at all.

Some intersections add to what a trajectory already carries. A tree on an exposed ridge records years of wind, each season bending what the last one left, and a sand spit is built from countless waves arriving at the same angle. Others replace what was carried: reprogramming factors wipe a cell's age-related marks (Section 3), and at fertilization almost everything is reset (above). Where they add up, what a trajectory carries changes how it meets the next one, and the dynamics of the system become complex through that accumulation. The history is in it, and it meets whatever comes next with that history.

5. Cascading Chaos

A cascade is a chain of intersections: one disturbance triggers the next, and that one the next. From outside, the chain reads as chaos. How a system is engineered — how tightly it couples its dependent parts across scales — decides whether a disturbance damps out or cascades.

Whether the chain can be traced back depends on the systems in it — how they are coupled and what imprint each leaves. Where they leave a readable trace, the cascade can be followed to where it began and corrected. Where they are coupled tightly enough that no trace survives as it runs, the cascade behaves as a genuinely chaotic failure, the engineered case where the coupling itself manufactures the chaos. What recovers is what had something in place to absorb or re-open the chain — slack, a circuit breaker, a revert path, a readable trace — and the trace is still read afterward, even when the cascade could not be predicted as it ran, which is where the next correction comes from. The claim is not that cascades are never chaos, but that their traceability depends on the systems, not on the appearance from outside.

A cascade's path is not set by the size of what triggers it. Disaster studies made this turn long ago: the hazard is only the trigger, and the root cause is vulnerability (Hewitt, 1983). What carries a disaster through a modern society is its dependency on networks, and where vulnerabilities overlap, escalation accelerates, and the secondary effects can become new sources of impact greater than the trigger (Pescaroli & Alexander, 2019). The cascade follows the lines of dependency, whatever the size of the shock.

Vulnerability can be hard to notice beforehand because, like friction, it shows itself only when a force sets it in motion; by the time it can be measured, it has already become impact (Pescaroli & Alexander, 2019). That is the problem and the method at once. You can read it afterward, from the imprint the cascade leaves — the 2003 blackout traced back to untrimmed trees in Ohio and a control-room alarm that had silently failed, two small vulnerabilities that overlapped into fifty million dark houses. Or you apply the force yourself, early, and read where it breaks before a real shock does.

The trigger can sit anywhere on the scale, and the cascade behaves the same. At Three Mile Island in 1979 it was a single valve stuck open, and the failure ran through the plant's tight couplings faster than the operators could read it. In the market crash of 2010 it was one oversized sell order, and the fall accelerated as the orders to buy drained away, until a five-second pause built into the exchange caught it and the price climbed back. At the far end sits a Carrington-class solar storm — a coronal mass ejection carrying, by one estimate, 150 billion times the energy of the Hiroshima bomb, of the kind that narrowly missed Earth in 2012 — which would do its damage not by its size but by finding every place our dependence on the grid and the satellites had gone unmapped. A stuck valve and a solar storm are orders of magnitude apart, yet each becomes a disaster the same way: through the dependencies already in place.

6. Forbiddances

Not every intersection revises a trajectory, and the paper does not claim it does; an intersection that leaves no imprint is not a counterexample to what follows.

Claim: In open systems, much of what is called chaos is shaped by intersections arriving from outside, including their cycles.
Forbiddance: An open system whose irregular behavior stays exactly as irregular once the intersections and cycles arriving from outside are taken into account.

Claim: An initial condition can shape a trajectory, but it is not a final destination; intersections along the way set new initial conditions.
Forbiddance: A real open system where the best possible knowledge of its starting state predicts where it ends up just as well as knowing what it met along the way.

Claim: A change of course at an intersection or collision can leave an imprint, and where that imprint is specific to its cause, the change can be traced back to it.
Forbiddance: A system that, once it can be measured, carries no readable trace of what it met along the way, only its initial condition.

Claim: The dynamics of a system become complex through accumulation at intersections.
Forbiddance: A system whose response is the same regardless of what it met before.

Claim: Whether a cascade can be traced back is set by how its systems are coupled and what each leaves behind, not by how chaotic it looks from outside.
Forbiddance: A cascade whose traceability follows from its outward appearance rather than from the coupling and the traces of the systems in it.

7. What This Paper Claims

Chaos theory is widely accepted as sensitivity to initial conditions. This paper takes it a step further and defines chaos as it really is in open systems: still being shaped along the way by what it meets.

Real complex systems are not determined in advance, because their course keeps being revised at intersections; what they carry can still show why they changed. The claim fails if a complex system's course turns out to be fully described by a deterministic chaotic model with no revision at intersections, or if different causes of a change leave traces that cannot be told apart. The mathematics of chaos stands. What is challenged is reading real systems as if they were as deterministic as the equations, a challenge with precedents in Popper's The Open Universe (1982) and Prigogine's The End of Certainty (1997).

What this makes possible is cause and effect that can be located: where a course was set, when the system was open to it, and the windows where a nudge would count. This is where prediction goes beyond storms and starts to understand the dynamics of what could go wrong before it does, and because that was traced, its correction is increasingly possible.

8. References

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