Flow

Prologue

1. 0:05Around 1961, Edward Lorenz, a meteorologist, ran a weather model on a computer, with twelve variables.
2. 0:12To rerun a stretch, he typed numbers back in from a printout. It showed three decimal places; the machine kept more. Say it held 0.506127: the paper said 0.506.
3. 0:26Same rules, nothing random. The two runs matched at first, then drifted apart: in the end, completely different weather.
4. 0:33What's on screen isn't his weather model, but three equations he published in 1963, from another problem: heated fluid rolling over. The same thing happens there.
5. 0:43The rules are exact, and short-term the runs agree; yet where a run ends up, long-term, can't be worked out. How can that be?
6. 0:51Answering it takes another view: not one number over time, but the whole state as one point, flowing through all possible states.
7. 0:59The shape of such a flow, without solving anything, shows where motion ends up, and what kind it is. And we'll see why, with a start known to finite precision, the far future slips out of reach.

IAn arrow at every point

1. 1:14What will a pendulum do next? Just two numbers decide: its angle, θ, and its angular velocity, p: how fast it turns, and which way.
2. 1:24Take them as coordinates: its state is a point on the plane. Each swing, there and back, takes the point once round; small swings make nearly an ellipse.
3. 1:33Together, its states form a phase space: a picture of states, not of the bob's path.
4. 1:38The equation doesn't spell out the motion. At each state, it gives two rates: the angle changes at rate p; p changes at a rate set by the angle.
5. 1:48A prime marks a rate of change. Together, the two rates give each point an arrow: which way to change, and how fast.
6. 1:55Across the plane, the arrows make a flow, like water. The comets show direction only.
7. 2:00The rule never changes with time: whenever a state comes round, it gets the same arrow. Following the arrows traces an orbit.
8. 2:08Short of the top, it swings back and forth: a closed loop. With enough energy, it goes over the top and keeps turning. On the dividing line, it has just enough to reach the top: slowing all the way, it never quite arrives.
9. 2:21Over the top, it leaves on the right, and comes back on the left, the same angle: each turn repeats the state.
10. 2:27So a state has one future, and orbits never cross.
11. 2:31Where the arrow is zero, the state stays put: a fixed point. There are just two: hanging down, and balanced upright.
12. 2:38Orbits circle the lower one. At the upper one, they come in from two sides, and leave along two others: nudge it, and it falls.
13. 2:45Four lines seem to meet there, yet nothing moving along them arrives: the point itself is an orbit, standing still.

IINear a fixed point

1. 2:56Close enough to a fixed point, the equation's curving hardly matters: each rate is, roughly, the offsets times fixed numbers, four in all. That's a linear rule.
2. 3:07Here, the four are 0, 1, −1 and 0. a + d is the trace; ad − bc, the determinant: here, 0 and 1.
3. 3:18Add friction, γ: the trace becomes −γ; the determinant stays 1.
4. 3:23Each γ is a separate rule, not friction growing over time.
5. 3:27Take a patch of states: the trace is how fast its area changes, in proportion to its size. Negative, the area shrinks and orbits close in; still turning, they spiral, and the swings die down.
6. 3:40Enough friction, and the turning stops. Along two lines, the arrows point along the line, with length in proportion to the distance. That signed ratio is an eigenvalue, λ: negative draws in, positive pushes out.
7. 3:54The two λs solve this quadratic: they sum to the trace, and multiply to the determinant. With a positive determinant, if τ² − 4Δ is positive, these lines exist, and there's no turning: a node; if negative, it spirals. The parabola is the border.
8. 4:13Back to the frictionless pendulum, at the upright point: the determinant is −1, so one λ is positive, one negative.
9. 4:21Drawn in along one line, pushed out along the other: every orbit leaves, unless it starts exactly on the line in. That's a saddle.
10. 4:30Back to the hanging point: trace 0, determinant positive. For the linear rule, that's a centre.
11. 4:36But it sits on a border, between closing in and pushing out: the dropped terms may act like friction, or like a push. Same linear part, different outcomes.
12. 4:46For the pendulum, it really is a centre, thanks to energy conservation.
13. 4:50This chart sums up linear rules: saddles, spirals, nodes and centres.
14. 4:55On a bright dashed line, for a real system, the linear part can't decide; and when a parameter crosses one, the motion nearby changes.
15. 5:03Under a linear rule, the trace also sets how area changes: negative, it shrinks; zero, unchanged; positive, it grows.

IIIA loop of its own choosing

1. 5:15A small swing without friction: x is the offset, y how fast it changes. Its orbits are a whole family of circles; any amplitude lasts, and none is special.
2. 5:26Nudge it, and it takes another amplitude, which lasts too.
3. 5:30In the 1920s, van der Pol, studying vacuum-tube circuits, wrote this: damping that depends on where it is now.
4. 5:37In the middle band, damping is negative, feeding energy in; outside, it's positive, draining it.
5. 5:44Near the origin, the trace is μ: for any μ above 0, the origin pushes out.
6. 5:49A small swing stays in the band, and grows. A big one is outside most of each turn, loses more, and shrinks. Near amplitude 2, over one turn, gain and loss balance.
7. 6:00And it can be proved: this is the only closed orbit; from anywhere but rest at the origin, motion approaches it.
8. 6:07The system, not the start, sets the amplitude: a stable limit cycle.
9. 6:12The stronger the feeding and draining, the less round the loop: a slow creep, a sudden jump, a creep, and a jump back. A relaxation oscillation: still perfectly regular.
10. 6:23The period grows too, spent almost all in the slow creeps. Van der Pol and van der Mark, in 1928, used such a circuit to model a heartbeat.
11. 6:33In a plane, a closed orbit is a wall: orbits can't cross, so inside stays in, and outside stays out.
12. 6:39Take a continuous flow on a plane, with a rule that never changes, and finitely many fixed points. Then, staying in bounds, an orbit can only approach one of three: a fixed point, a closed orbit, or a loop with fixed points. That's the Poincaré–Bendixson theorem.
13. 6:57Now put it to work: near the origin, the flow pushes out; an outer orbit comes back further in. Between them, a band that nothing leaves, with no fixed point: so, without solving anything, a closed orbit must lie inside.
14. 7:11Pulling neighbours apart, then folding them back, again and again, in bounds, is beyond such a flow. A third dimension, a force that changes with time, or a step-by-step rule makes it possible.

IVThe kicked pendulum

1. 7:27Without friction, the pendulum keeps its energy. Angle and angular velocity set it, and each orbit holds its own value: a contour line of energy.
2. 7:36Because energy is conserved, orbits around the hanging point close up: a true centre.
3. 7:41Release a blob of neighbouring states. Each energy goes round at its own pace, so the blob is slowly stretched, and wound into a thread.
4. 7:49But however much it narrows one way, it stretches as much the other: the shape changes; the area doesn't. That's Liouville's theorem, for systems like this pendulum.
5. 8:00Since its area can't shrink, the blob can't be drawn into a point, or pressed onto a loop. Van der Pol's oscillator, last chapter, does gather states onto its cycle.
6. 8:10Now change the pendulum: instead of gravity's steady pull, a sharp kick at regular intervals, with free turning in between. Each kick's strength is set by the angle, at that moment.
7. 8:21Look only just before each kick, and each state gives the next: a step-by-step rule, the standard map. Energy is no longer conserved, so points aren't tied to energy contours. They're snapshots, not a line: “no crossing” doesn't apply. The extra dimension: time till the next kick.
8. 8:40Kick harder, and a layer forms by the dividing line: two near-identical starts, after a few dozen steps, go separate ways.
9. 8:48Staying in bounds, never settling or repeating, with neighbours flying apart: that's chaos. It's still deterministic, and it still keeps area.
10. 8:56These long curves are over-the-top orbits, bent by the kicks but unbroken: with gentle kicks, most survive. That's the KAM theorem.
11. 9:06Kick harder, and they break one by one. Those turning, on average, a simple fraction of a turn per kick, break first, into island chains. The harder a rate is to approximate, the sturdier its wall.
12. 9:18Hardest of all: rates tied to the golden ratio. That wall breaks last. Until then, points hop along it, never repeating, never getting off; it attracts nothing, but like a wall, keeps the two sides apart.
13. 9:32Numerical studies put its breaking point, for this map, at K of about 0.9716. Beyond it, chaotic points can slowly wander far, and come back; and in the chaotic sea, regular islands remain.

VA strange attractor

1. 9:49In 1963, Lorenz took a fluid heated from below, rolling over, and cut its equations to three variables.
2. 9:57Together, they're a point in three dimensions: scaled numbers, not a place in the fluid. Again, the equations give every point an arrow.
3. 10:05It's no faithful model of convection; yet equations this simple, and exact, can produce chaos.
4. 10:11All three fixed points are unstable: the origin is a saddle, the other two spiral outward; and there's no stable loop either.
5. 10:19This orbit stays in bounds, yet never stops or closes up: a few turns on one side, a jump across, and back, in no repeating pattern.
6. 10:27A continuous flow on a plane, with an unchanging rule, has no room for this. But here there's a third variable: strands that seem to cross, in fact, pass one behind the other.
7. 10:38This is the prologue's answer: from two almost identical starts, the gap doubles, on average, about every three-quarters of a time unit: exponential growth. Its rate, λ, about 0.906, is a growth rate, like chapter II's eigenvalues; here, an average.
8. 10:57Each extra decimal at the start, on average, buys only a fixed stretch of forecast. Short-term, it's accurate; but with finite precision, the gap keeps growing, until it's as big as the butterfly.
9. 11:08The rules are exact, with nothing random; what hides the far future is this: sensitivity to the start.
10. 11:15Now a blob of states. The system dissipates: every time unit, any blob shrinks in volume, about a millionfold.
11. 11:22Yet it doesn't shrink to a point; it spreads over both wings. Almost every start ends up on this shape: an attractor.
12. 11:30It's stretched one way, squeezed harder another, and folded back, into endless thin sheets; from how fast it stretches and squeezes, its dimension comes out at about 2.06. Layered, and sensitive: a strange attractor.
13. 11:45Lorenz also tried something else: noting only the peaks of z, each time it reaches a high point.
14. 11:51Pair each peak with the next, the first across, the second up: one point.
15. 11:55The points fall on a thin, tent-shaped band: from one peak, you can roughly work out the next.
16. 12:01The steep rise stretches differences; past the top, they're folded back. Stretch and fold: one root of chaos.
17. 12:09Next, an even simpler rule of this kind, not derived from Lorenz's equations.

VIOne road to chaos

1. 12:17One of the simplest: the next number is r times x, times one minus x. The ecologist Robert May used it, as a simple population model.
2. 12:26Like the tent, it rises and falls, stretching and folding, but with a round top. And step by step, “no crossing” doesn't apply: one variable is enough for chaos.
3. 12:37For r between 1 and 3, from anywhere between 0 and 1, it approaches the same fixed point.
4. 12:43Each column is one r, run for a long time, showing only where it ends up. The horizontal axis isn't time.
5. 12:50Past r = 3, the fixed point loses stability: the sequence hops between two values.
6. 12:57A little further, and two become four: these four lines are one cycle, visited in turn.
7. 13:02Then eight, sixteen, doubling again and again. Divide each gap by the next, and the ratio approaches 4.669: a constant called Feigenbaum's δ.
8. 13:13So before r reaches about 3.5699, it has doubled infinitely many times.
9. 13:19Past this dashed line, in many places, the sequence never settles into a cycle: it never repeats, and nearby starts part exponentially, the butterfly's kind of chaos.
10. 13:29Yet chaos has windows of order: past r = 1 + √8, three values take turns again, neatly.
11. 13:36Further on, the window doubles again, six, twelve, and then falls back into chaos.
12. 13:41Zoom in where the doublings pile up, and the same doubling appears again: the closer you look, the more it looks like the whole.
13. 13:48Feigenbaum found, and Coullet and Tresser independently: take any one-humped curve; if its top is round like a parabola's, not sharp, and no flatter, the ratio also tends to 4.669.
14. 14:01Experiments saw the first doublings too, with measured ratios between 4.3 and 4.7.
15. 14:07This is one road to chaos, of several; along it, one constant holds, for a whole class of systems.

Coda

1. 14:15For continuous motion, an equation gives every state an arrow. Without solving anything, the arrows show where motion ends up: in a point, on a loop, or never repeating, on a strange attractor; those are the kinds we've seen.
2. 14:29In chaos, the rules are still exact: short-term, we can forecast, and we know it's on the butterfly; but with finite precision, where it'll be far ahead, we can't say.
3. 14:40Add a dimension, a kick now and then, or a step-by-step rule, and chaos becomes possible.
4. 14:45And on one road to chaos, one constant holds, for a whole class of systems.
5. 14:50But a continuous flow on a plane, with an unchanging rule, has no room for chaos.
6. 14:55An arrow at every point; together, a flow.