# Chaos

One mathematical topic that I find fascinating, but which I’ve never had a chance to study formally is chaos. I’ve been sort of non-motivated about blog-writing lately due to so many demands on my time, which has left me feeling somewhat guilty towards those of you who follow this blog. So I decided to take this topic about which I know very little, and use the blog as an excuse to learn something about it. That gives you something interesting to read, and it gives me something to motivate me to write.

I’ll start off with a non-mathematical reason for why it interests me. Chaos is a very simple idea with very complex implications. The simplicity of the concept makes it incredibly ripe for idiots like Michael Crichton to believe that he understands it, even though he doesn’t have a clue. There’s an astonishingly huge quantity of totally bogus rubbish out there, where the authors are clueless folks who sincerely believe that their stuff is based on chaos theory – because they’ve heard the basic idea, and believed that they understood it. It’s a wonderful example of my old mantra: the worst math is no math. If you take a simple mathematical concept, and render it into informal non-mathematical words, and then try to reason from the informal stuff, what you get is garbage.

So, speaking mathematically, what is chaos?

To paraphrase something my book-editor recently mentioned: in math, the vast majority of everything is bad. Most functions are non-continuous. Most topologies are non-differentiable. Most numbers are irrational. Most irrational numbers are undescribable. And most complex systems are completely unstable.

Modern math students have, to a great degree, internalized this basic idea. We pretty much expect badness, so the implications of badness don’t surprise us. We’ve grown up mathematically knowing that there are many, many interesting things that we would really like to be able to do, but that can’t be done. That realization, that there are things that we can’t even hope to do, is a huge change from the historical attitudes of mathematicians and scientists – and it’s a very recent one. A hundred years ago, people believed that we could work out simple, predictable, closed-form solutions to all interesting mathematical problems. They expected that it might be very difficult to find a solution; and they expected that it might take a very long time, but they believed that it was always possible.

For one example that has a major influence on the study of chaos: John Von Neumann believed that he could build a nearly perfect weather prediction computer: it was just a matter of collecting enough data, and figuring out the right equations. In fact, he expected to be able to do more than that: he expected to be able to control the weather. He thought that the weather was likely to be a system where there were unstable points, and that by introducing small changes at the unstable points, that weather managers would be able to push the weather in a desired direction.

Of course, Von Neumann knew that you could never gather enough data to perfectly predict the weather. But most systems that people had studied could be approximated. If you could get measurements that were correct to within, say, 0.1%, you could use those measurements to make predictions that would be extremely close to correct – within some multiple of the precision of the basic measurements. Small measurement errors would mean small changes in the results of a prediction. So using reasonably precise but far from exact or complete measurements, you could make very accurate predictions.

For example, people studied the motion of the planets. Using the kinds of measurements that we can make using fairly crude instruments, people have been able to predict solar eclipses with great precision for hundreds of years. With better precision, measuring only the positions of the planets, we can predict all of the eclipses and alignments for the next thousand years – even though the computations will leave out the effects of everything but the 8 main planets and the sun.

Mathematicians largely assumed that most real systems would be similar: once you worked out what was involved, what equations described the system you wanted to study, you could predict that system with arbitrary precision, provided you could collect enough data.

Unfortunately, reality isn’t anywhere near that simple.

Our universe is effectively finite – so many of the places where things break seem like they shouldn’t affect us. There are no irrational numbers in real experience. Nothing that we can observe has a property whose value is an indescribable number. But even simple things break down.

Many complex systems have the property that they’re easy to describe – but where small changes have large effects. That’s the basic idea of chaos theory: that in complex dynamical systems, making a minute change to a measurement will produce huge, dramatic changes after a relatively short period of time.

For example, we compute weather predictions with the Navier-Stokes equations. N-S are a relatively simple set of equations that describe how fluids move and interact. We don’t have a closed-form solution to the N-S equations – meaning that given a particular point in a system, we can’t compute the way fluid will flow around it without also computing separately how fluid will flow around the points close to it, and we can’t compute those without computing the points around them, and so on.

So when we make weather predictions, we create a massive grid of points, and use the N-S equations to predict flows at every single point. Then we use the aggregate of that to make weather predictions. Using this, short-term predictions are generally pretty good towards the center of the grid.

But if you try to extend the predictions out in time, what you find is that they become unstable. Make a tiny, tiny change – alter the flow at one point by 1% – and suddenly, the prediction for the weather a week later is dramatically different. A difference of one percent in one out of a million cells can, over the space of a month, be responsible for the difference between a beautiful calm summer day and a cat-5 hurricane.

That basic bit is called sensitivity to initial conditions is a big part of what defines chaos – but it’s only a part. And that’s where the crackpots go wrong. Just understanding the sensitivity and divergence isn’t enough to define chaos – but to people like Crichton, understanding that piece is understanding the whole thing.

To really get the full picture, you need to dive in to topology. Chaotic systems have a property called topological mixing. Topological mixing is an idea which isn’t too complex informally, but which can take a lot of effort to describe and explain formally. The basic idea of it is that no matter where you start in a space, given enough time, you can wind up anywhere at all.

To get that notion formally, you need to look at the phase space of the system. You can define a dynamical system using a topological space called the phase space of the system. Speaking very loosely, the phase space P of a system is a topological space of points where each point p∈P corresponds to one possible state of the system, and the topological neighborhoods of p are the system states reachable on some path from p.

So – image that you have a neighborhood of points, G in a phase space. From each point in G, you traverse all possible forward paths through the phase space. At any given moment t, G will have evolved to form a new neighborhood of points, Gt. For the phase space to be chaotic, it has to have the property that for any arbitrary pair of neighborhoods G and H in the space, no matter how small they are, no matter how far apart they are, there will be a time t such that Gt and Ht will overlap.

But sensitivity to initial conditions and topological mixing together still aren’t sufficient to define chaos.

Chaotic systems must also have a property called dense periodic orbits. What that means is that Chaotic systems are approximately cyclic in a particular way. The phase space has the property that if the system passes through a point p in the neighborhood P, then after some finite period of time, the system will pass through another point in P. That’s not to say that it will repeat exactly: if it did, then you would have a repeating system, which would not be chaotic! But it will come arbitrarily close to repeating. And that almost-repetition has to have a specific property: the union of the set of all of those almost-cyclic paths must be equivalent to the entire phase-space itself. (We say, again speaking very loosely, that the set of almost-cycles is dense in the phase space.)

That’s complicated stuff. Don’t worry if you don’t understand it yet. It’ll take a lot of posts to even get close to making that comprehensible. But that’s what chaos really is: a dynamical system with all three properties: sensitivity to initial conditions, overlaps in neighborhood evolution, and dense periodic orbits.

In subsequent posts, I’ll spend some more time talking about each of the three key properties, and showing you examples of interesting chaotic systems.

## 0 thoughts on “Chaos”

1. John Armstrong

Jim Yorke explained topological mixing by pointing to the horseshoe map. Take a square and stretch it out into a long, thin rectangle. Then fold it into a horseshoe and lay it across the original square so that the bend in the horseshoe is outside on one side of the square and the two ends of the horseshoe are outside on the other side. Then iterate this map.
It turns out that a very large number of chaotic dynamical systems actually have a horseshoe hidden in them somewhere — a region that is stretched (giving sensitive dependence) and folded (bringing far-separated points near) and then laid across itself (mixing).

2. phed

Not really a big thing, but perhaps your mention of Crichton should be in past tense because he died last November.

3. Lyle Brecht

Do you have any experience w/ PRA (probabilistic risk assessment). They use this methodology for calculating risks associated w/ complex engineered systems e.g. nuclear power plants, etc. What I am thinking is that these PRA apples linear equations to assess risk to complex, non-linear systems that would exhibit chaotic behavior. I am just wondering if any mathematical work has been done w/re to chaos and the use of PRA methods?

4. Brian Utterback

Look, I certainly don’t agree with everything Michael Crichton has said, or the political positions he took or even his application of science, but he was a great writer, at least until his story telling started taking a back seat to the his political views. The fact that his understanding of advanced mathematics and technology was imperfect is hardly a reason to call him an idiot. After all, you even say his understanding is partial, which is better than most people who don’t have even that level of understanding. If you want to criticize his treatment of chaos, fine, tell us what he got wrong. But name calling isn’t required, especially since you admit your own understanding was incomplete prior to writing on this topic.
Otherwise, a great post.

5. graveerror

I stopped reading your post and deleted your feed from my rss reader as soon as you called Michael Crichton an idiot for no apparent reason. Instead of chaos, maybe you should study manners and humility.

6. Nathan

Very nice post. I look forward to the sequels, this kind of thing is far more interesting to me than data structures π

7. Queef

It’s funny that some people above are offended that you pointed out the fact that Michael Crichton was a subpar writer. I was expecting talk of Chaos theory.

8. mos

Stick me in the “what’s he got against Michael Crichton?” group. The guy wasn’t a mathematician, he was an author — an entertainer. He wrote to the common man, who has no grasp of any real math, and wouldn’t be interested in the blog post I just read.
I thought it was a good read except for those times when I was distracted away from the main point.

9. oscar zoalaster

Re: Michael Crichton: “The fact that his understanding of advanced mathematics and technology was imperfect is hardly a reason to call him an idiot.
If that was the case you would have an excellent point. Unfortunately Michael Crichton willfully had – and articulated to a large audience – imperfect understandings of mathematics and technology. His only interest in mathematics and technology was whether he had the technobabble for some ‘cool’ plot idea.
Any writer, no matter how otherwise skilled, is an idiot if he/she writes to the ‘rule of cool’. And anyone who doesn’t make the effort to educate him/herself about the mathematics and technology he/she is ‘talking to millions’ about is an idiot.

10. Richard

I’ve been looking into it, and I’m starting to suspect that A LOT of things Michael Crichton wrote weren’t true at all. It’s almost like he was writing fiction.

11. Oliver X

If Michael Chrichton were merely a mediocre science fiction author, then we wouldn’t be talking about him here. But he was more than that: he was an outspoken critic of solid environmental, health, and climate science in the mainstream media. And that makes him fair game for criticism himself.

12. foobeans

Chaos theory is where the question of free will vs determinism breaks down. When in a deterministic but chaotic, dynamic system the question of free will becomes meaningless.

13. haig

I don’t think Mark would call Crichton an idiot if it were only for Crichton’s anti-science stance in almost all his fictitious works. No, I think the harsh words are more directed towards Crichton’s public actions outside his fiction, specifically his anti global-warming propaganda that used not well understood technical ideas (including chaos/complex systems science) to supposedly disprove anthropogenic climate change. He basically used the same ‘bad math’ thinking that the ID et al idiots use to further their ideologies, hence lumping him into that category of ‘idiot’ seems slightly justified.
I personally don’t think he was an idiot, I liked a lot of his works (when I was 12) and think he was a decent human being, he just might have taken the anti-science devil’s advocate role too far with certain issues.

14. Scott McKuen

Mark,
Getting away from both the actual topic and what everybody appears to want to talk about (chaos and Michael Crichton, in some order), you make this assertion:
There are no irrational numbers in real experience.
Not intending to offend you, but to me this sounds like the kinds of overly-simplistic statement about the relationship between math and the real world that you would usually attack pretty fiercely.
Are you just being glib here?

15. Michael Ralston

Scott @15: The statement may be a little bit glib, but it’s also true – every measurement only provides a finite supply of information, thus is representable by a rational number.
Nothing we can do can elucidate an irrational measurement, and for everything we do in reality a rational approximation to any irrationals will suffice.
For instance, I believe I recall reading that 39 digits of Pi are sufficient to approximate the circumference/radius ratio of a circle the size of the entire universe to within the diameter of an atom.
Certainly any normal experience with circles or triangles or other such entities is handleable with rationals. (Even, perhaps, to not-so-exact degrees of precision, as witness by the Biblical passage that would indicate Pi=3 if the numbers are taken to be completely precise.)

16. John

Michael Crighton said climate was unpredictable and that we couldn’t do anything to change it.
Mark CC, in his post about chaos, said that weather was unpredictable and that we couldn’t do anything to change it.
Two idiots, or none, methinks.

17. John

@csrster Very funny. I see what you did there.
I’ve yet to hear an explanation of the certainty of climate predictions weighed against the chaotic nature of the weather.
I guess I’m gonna have to keep waiting.

18. Frank Shearar

@John, #17: Mark didn’t say you couldn’t control weather.
In fact, it’s rather easy to affect weather: google “cloud seeding”, for instance.
Of course, it’s a corollary to the post that you can’t predict HOW you affect the weather.

19. Scott McKuen

@Michael, 16: Sure, the density of the rationals lets us approximate everything within measurement error. The density argument applies equally to the dyadic rationals (the denominator can only be a power of 2). You never hear anybody suggest that we should think of the world as “really” just the finite-days part of the surreal numbers, even though that’s exactly what the dyadic rationals are.
The surreals are just a model based on finite sets of integers, exactly like the ordered-pair perspective of the rational numbers is. What’s the difference from a physicist’s perspective? It would be really annoying to talk about the charge of the quark without believing in our “real experience” of the number 1/3.
This isn’t my brief for the surreals as being “the real world” instead of the rationals. Rather I just don’t agree that the square root of two is somehow less “real” by whatever standard than a rational number with denominator > 10^30.

20. Blake Stacey

You say weather is “chaotic”, but I’ve noticed that here in Massachusetts, snow is more likely on Christmas than on the Fourth of July. Explain that with your precious “science”!!!

21. Oscar

First of all, I’ve read a couple of (non-mathematical) books about chaos theory, and I can honestly say I think I’ve learned more from your post than from those books, which is really awesome π

In 1990, Crichton published the novel Jurassic Park. Crichton utilized the presentation of “fiction as fact”, used in his previous novels, Eaters of the Dead and The Andromeda Strain, in conjunction the mathematical concept of chaos theory and its philosophical implications to explain the collapse of an amusement park […]

I think it is a mistake from the *reader’s* part to assume that statements made in a work that’s clearly fiction to be taken as fact. But I’ve never read the JP books, so I ask: in those books, does the author overtly state that what he says about chaos theory is correct?

22. Blake Stacey

It’s been years since I even picked up Jurassic Park, but I have a moderately good memory for afterwords, and I do recall an “author’s note” or something of the sort appended to the book in which Crichton said, basically, “ZOMG CHAOS THEORY IZ REALZ0RS!”

23. Chris W.

John:
Think of a pot of water boiling on a stove. Predicting the weather from day to day is like trying to predict where the next five bubbles will arise and how big they are going to be. Predicting the climate over the next few decades is like predicting what will happen to the average size and number of bubbles if you turn the burner up or down.
One of these tasks is much easier than they other.

24. Paul W. Homer

Is this a real example of Chaos Theory? A small mention of a famous author in a negative context, radically skewing all of the comments towards that topic and not the original blog one?
That seems to work for the first two properties, but what about dense periodic orbits? Perhaps you need a follow up post π
Paul.

25. Mark C. Chu-Carroll

Re lots of folks:
I’m sorry if you’re offended by my characterization of Crichton, but I stand by it.
Yes, he was writing fiction. I don’t dispute that.
But he frequently set up characters in his fiction as authoritative lecturers, giving detailed explanations of some phenomenon that was important to his plot. And those explanations were frequently total, utter bullshit.
With respect to Chaos theory, Crichton created characters that he described as experts, and had them give long lectures on chaos theory and what it meant – only he got it completely, utterly wrong.
That pisses me off. And not just Crichton, but many authors who did the same thing. Heinlein’s juvenile novels were frequent, terrible offenders in this area. He’d pick up some little bit of information, and believe that he understood it, and use it to launch into long-winded bullshit that was completely wrong. (For example, I remember one of his where the main character is talking to a Wise Old Man Engineer in a spaceship, asking about why they couldn’t go faster than light. Heinlein knew that you couldn’t go faster than light, and he knew it had something to do with Einstein, so he had the engineer babble a bit – but he didn’t know anything about *why* you couldn’t, or what would happen as you accelerated. So he just made it up.)
I’ve also seen Crichton interviewed around the time the Jurassic Park movie came out, and he used the interview as a platform for spouting yet more nonsense. The guy took the one-sentence description of “the butterfly effect”, and believed that by understanding that, that he understood chaos theory; and that because he spent a lot of time thinking about the butterfly effect and how it could describe various phenomena, that he was an expert on how to apply chaos theory to the real world. He really believed that he *knew* it, and that he was qualified to lecture on it. And that’s the problem to me: knowing a tiny little corner of something, and concluding that you’re an expert.
Obviously, I have a lot of disagreements with what Crichton believed politically. But that’s really, seriously irrelevant. I mentioned Heinlein just two paragraph above: I disagree completely with just about everything that Heinlein wrote about politics. But I *love* many of his books. He was a brilliant storyteller, and the fact that I don’t agree with the political arguments that he embedded in many of his books doesn’t change that one whit.
Crichton, on the other hand, I think was a lousy hack. I think that as a writer, he wrote incredibly pedestrian prose; his plots were predictable; his characters were dreadful. And he constantly included dreadful lectures on science that were just *wrong*.

26. Paul W. Homer

Of course you have a right to your opinions Mark, but we should all be careful about how strictly we judge people based on what they know or do not.
There are few humans for which there isn’t a perspective on their efforts that doesn’t show them up to be wrong about something. Being human, we are all *wrong* sometimes. Intentionally, accidentally or historically. We only ever see a small slice of the world around us. We’re forced to make our way through with incomplete information, it’s the nature of our existence.
If Crichton set out to entertain, then he accomplished that quite well. Lots of people like his work. We shouldn’t be too hard on him if the demands of his public lead him to to try to crossover to a more serious stance. It could happen to any of us π
Paul.

27. Paul Carpenter

Chaos theory has lots of pretty pictures and poetic analogies, which will always attract laymen who are more interested in talking about maths than doing or understanding it. It’s a field that I’d like to study when I have the time – even I’m slightly attracted by the pretty pictures.
Question: if ill-conditioning is part of what defines chaos – then does that mean that chaos isn’t an absolute state, since there is no line between a well conditioned system and an ill conditioned one?

28. John

@Chris W
Very good analogy, but we built the kettle and understand everything necessary about it, and predicting that it will heat up is a trifle when we designed it to do just that. The climate is neither built nor fully understood by us, and there’s no way we can be sure that the macro-behaviour is any more or less predictable than the unpredictable micro-behaviour.
It’s really disappointing to hear Mark attacking Crighton, when there’s so much more innaccurate, idiotic and misinterpreted figures in what passes for climate science today.

29. Dan L.

I’ve yet to hear an explanation of the certainty of climate predictions weighed against the chaotic nature of the weather.
I guess I’m gonna have to keep waiting.
Posted by: John | June 8, 2009 7:25 AM

I don’t think you have a good grasp of the relationship between weather and climate. Weather is a highly variable chaotic phenomenon, but we wouldn’t be able to talk very coherently if it didn’t show certain patterns. These patterns should be pretty obvious; snow is extremely rare in Florida and (somewhat counterintuitively) Antarctica. While common in the northeast US, it is almost unheard of in June or July whereas it is quite uncommon to go through January without at least a dusting. These patterns are often wholly predictable and not the result of a chaotic system — seasons, for example, are dependent on the non-chaotic mechanics of the earth revolving around the sun with an inclined axis of rotation. Determining the hottest and coldest (on average) times of the year for any particular location on earth does not require any analysis of chaotic systems.
Climate is basically exactly those parts of weather that are predictable and are not chaotic. This is a rule of thumb, though, not a definitive statement.

30. Dan L.

It’s really disappointing to hear Mark attacking Crighton, when there’s so much more innaccurate, idiotic and misinterpreted figures in what passes for climate science today.

I’m sorry, but you’re giving me the impression that you don’t know enough about it to be able to make a statement like this without talking out of your ass. Can you back it up?

31. James

Dense periodic orbits and topological mixing imply sensitivity to initial conditions don’t they?

32. ChrisZ

Quick question, is there a separate field of study devoted to dynamical systems that just exhibit one of the three properties mentioned (like sensitivity to initial conditions, for example)? And if not, why not?

33. Mark C. Chu-Carroll

re #37:
I don’t know of anything that specifically looks at dynamical systems that are sensitive to initial conditions. My guess would be that people who study dynamical systems care about that, but that I’d doubt it’s a separate field of study. In general, you need some kind of strong common structure to create something interesting enough to create a new field of study. Chaos has this collection of properties that create something uniquely interesting. Take away some of it, and you lose that basic structure.
Particularly with respect to sensitivity: it’s hard to imagine a whole field of study based on that, because so many simple and uninteresting functions are extremely sensitive to initial conditions.
Pretty much every source I’ve looked at about chaos includes some form of this as an example: think of the function y=x^2. If you start with x=0.99 and iterate, then you’ll have a function which decreases monotonically towards zero. If you start with y=1.01, you’ll have a function which monotonically increases towards infinity. If you start with 2 and 4, after 5 steps, starting from 2, you’ll have 4294967296; starting from 4, you’ll have 1.8×1019. Pretty different: it’s pretty sensitive to initial conditions. And the further out you go in time, the further they’ll diverge.
But there’s nothing chaotic about it. And frankly, there’s really nothing interesting about the difference in result iterating y=x2 from different starting points.

34. Mark C. Chu-Carroll

Re #33:
I’m an equal opportunity critic. If you have an actual good example of bad math in climate science, I’ll be glad to rip it to shreds. But “bad math” and “John doesn’t agree with the math” are two quite different things. You seem to want to basically take conclusions that you don’t want to agree with, and argue that the chaotic nature of weather means that they must inevitably be wrong – or at least that they must be treated as so uncertain that they can’t be used for any kind of future planning.
That’s nonsense.
Let me give you a nice little metaphor.
I save money for the day in the hopefully distant future when I retire. I invest it in several different ways. In the short term, the value of my retirement investments on any given day is incredibly unpredictable. In fact, it’s downright chaotic. There’s no way to make any meaningful prediction about what my investments will be worth next tuesday, or on july 19th.
But I can predict that the value of my investments will grow over time. I can’t be 100% certain – nothing is every 100% certain – but I can safely predict that my retirement investments will be dramatically larger 20 years from now than they are today.
What you seem to want to do is to argue that there’s no way to know what my retirement investments will be worth, because it’s based on an inherently chaotic system, and that therefore, anyone who talks about long-term growth expectations is talking out their ass.

35. Warren

Michael Crichton: the thinking man’s Stephenie Meyer.
Posted by: Blake Stacey | June 7, 2009 11:05 PM

…and

[Crichton] frequently set up characters in his fiction as authoritative lecturers, giving detailed explanations of some phenomenon that was important to his plot. And those explanations were frequently total, utter bullshit.
Posted by: Mark C. Chu-Carroll | June 8, 2009 10:27 AM

…thus me:
Michael Crichton: The thinking man’s Ayn Rand.

36. Blake Stacey

Warren:
A hit! A very palpable hit!
This discussion reminded me of something I read a while back: Alexei Panshin’s 1969 review of The Andromeda Strain.

[The Andromeda Strain] has been favorably reviewed by Life, Look and the New York Times, and it has sold to the movies for an impressive sum. It is also cheap, sensationalistic, hastily written trash…. The story is either a plausible thriller—that is, you believe in the plague and in the efforts of the scientific and medical team to cope with it—or it is nothing.
Crichton bolsters his story with easy expertise and massive documentation, but the story never hangs together. The main reason is that Crichton invents his story as he goes along and is satisfied to put down the first thing that comes to mind, and one lie contradicts the next. Thus you have a bacteriologist who has won the Nobel Prize for work done in his spare time as a law student (Crichton consistently oversells)—but who doesn’t know that he has a vein in his wrist…. Thus you have an Army van with a rotating antenna on top tacking back and forth across the Mojave desert taking triangulations every twenty miles on a grounded satelliteβthe landing site of which has already been predicted with an error of a few hundred yards. Two vans, we are told, would be suspicious. Thus you have a portentous scientific report on the probability of contact between man and other life forms with all figures to four places and a list of possibilities of encountering a life form more advanced than our own (the “7 +” level of data handling, if you please), or the possibility of encountering a life form radically different from our own, or the possibility of encountering no life at all. Crichton’s documentation is fake, his expertise is false, and even his basic problem turns out to be a fraud—after a few days the plague ups and goes away.

(Quoted from Zeno’s place.)

37. Ergo Ratio

Re: 37
I did my graduate work studying the complex dynamics of bursting neurons, and I only concerned myself with the dependence of the system on initial conditions. However, I wasn’t studying, in the abstract, dynamics in general under such constraints.

38. Stephen Wells

@”John”: consider the difference between predicting the next spin of a roulette wheel, and predicting whether the house will make a profit today.

39. MJ

Meine Gott, you idiots babbling loudly into the breeze. Why aren’t you out helping the homeless in your areas? Are we all too lazy?

40. MattR

Michael Crichton: the thinking man’s Stephenie Meyer.
Posted by: Blake Stacey | June 7, 2009 11:05 PM

Michael Crichton: almost as pedestrian as Dan Brown.

41. Paul Murray

But the requirement for dense periodic orbits – can’t you satisfy that by just takinhg those parts of the phase space where the orbits happen to be dense, and saying “this is the chaotic system that I’m talking about – ignore that man behind the curtain”?
I hope we get examples of sytems that satisfy two conditions but fail the third, and hence are not chaotic. Damped oscillation, perhaps, is not dense, and thermal motion not periodic.

42. ChrisZ

Thanks, Mark, for responding to my question, I think your response actually helped me a lot with understanding what it means to be chaotic (with the understanding that helping me a lot still hasn’t gotten me very far). I wish I had taken more math in college.
Also, a thanks to Ergo Ratio for responding as well.

43. cm

Mark,

But I can predict that the value of my investments will grow over time. I can’t be 100% certain – nothing is every 100% certain – but I can safely predict that my retirement investments will be dramatically larger 20 years from now than they are today.

Sincerely, how can you know that?

44. koko

While it is an admitedly poor analogy, one could think of weather and climate in terms of turbulent flow. Weather is the individual turbulent structures giving an instantaneous detailed picture, while climate is the bulk properties of the flow. One does not need to know the exact instantaneous details of the structures to predict bulk flow behavior. Similarly, it is much easier to predict the bulk properties of the environment (climate), than it is to predict the day to day, or hour to hour flucuations (weather).

45. Dan L.

Meine Gott, you idiots babbling loudly into the breeze. Why aren’t you out helping the homeless in your areas? Are we all too lazy?

That’s easy. I am the homeless in my area. Why are you posting here instead of going out and helping me?

46. SLC

Re Michael Crichton
We should recall that former President George W. Bush consulted the late Dr. Crichton on the subject of global climate change. Of course, the fact that Dr. Crichton had not a shred of expertise on the subject doesn’t seem to have bothered the former president or the global warming deniers.

47. Doug

Hi Mark,
I may have misunderstood your comments in #38 re #37, but an excellent book to start learning mathematical dynamics is:
Boris Hasselblatt, AB Katok, A first course in dynamics: with a recent panorama of developments, Cambridge University Press, 2003.
Mathematical Dynamics is closely related to Ergodic Theory [AMS 2000 37-xx]. The primary difference seems to be that the latter uses 2D planar surfaces with a diagonal while the former uses 3D cylndrical surfaces with a helix.

48. John_in_Oz

There’s no subtext in what follows; I’m certain you are not making stuff up. I’m troubled by the fact I’ve never heard of your second and third conditions for a system to be chaotic.
They don’t seem to be mentioned in the Wikipedia article on the subject either.
I’ve read James Gleick’s ‘Chaos’. I’ve read many ‘popularised science’ books, several of which include sections on Chaos theory.
I’m wondering what your sources are, and I’d love a reasonbable explanation as to why these conditions aren’t mentioned in most popularisations.
Thanks.
Apropos of Heinlein’s work, I’ve heard he was renowned among his peers, including Asimov, for keeping as up to date with science as was humanly possible at the time. If he was wrong, I (and his other fans, i assume) would appreciate knowing which book and what errors. He certainly did have some terrific authorial techniques for implying that the characters knew more than the science of the fifties did, but to this day I know of no error that he made that could have been recognised at the time.

49. John_in_Oz

Arrgh; I was wrong, dammit- they ARE mentioned in Wikipedia. *Blush*.
Not the other sources, though. (I hope!)

50. Dan L.

Apropos of Heinlein’s work, I’ve heard he was renowned among his peers, including Asimov, for keeping as up to date with science as was humanly possible at the time. If he was wrong, I (and his other fans, i assume) would appreciate knowing which book and what errors.

I haven’t read a whole lot of Heinlein, but I agree with this. “Have Space Suit Will Travel,” although admittedly juvenile in many respects, actually tries to address pragmatic issues involving space suits. I couldn’t find anything terribly wrong with the science in “Starship Troopers” other than the fact that giant insects are probably only possible on low gravity planets (and if I recall correctly, most of the action takes place on a relatively high gravity planet).
I can see what you’re (Mark) saying about his politics, although I think that’s a little more complex than you allow; compare the political vision from “Stranger in a Strange Land” to that from “Starship Troopers” and you’ll get an idea of what I mean.
That said, I’ve never worried too much about the science in science fiction, though I tend to enjoy SF more the more accurate the science is. I don’t think futurism or prediction is really the goal of science fiction. The best science fiction, in my opinion, define worlds in which we can carry out thought experiments regarding human ethics.
A good example might be Wells’ “The Invisible Man.” Clearly, the science is not very good (though he does provide a pretty good chunk on optics to try to justify it, so not the worst book in this respect either). The point of the book, however, is really to discuss the consequences of technology giving some human beings power over others and of the impact of that power on the humanity of the oppressor and the oppressed.
Philip Dick’s whole catalog is along these lines — bad science, but very innovative thought experiments for probing what it means to be a human being in the first place.
I haven’t read any Crichton so I can’t really say whether or not he would be a good sci fi writer by this criterion.

51. llewelly

Uh oh. The Civility Police are on patrol. Be nice or they’ll stalk off in a huff.

52. j a higginbotham

1) #39, 49 How can I transfer mine from TIAA/CREF to Mark? I’m sure that what I have now isn’t dramatically different from what I put in, starting in the 90’s.
2) Ignoring that I don’t understand the math, what are the necessary conditions for the weather to be unpredictable in the sense that arbitrarily small changes at one time will result in dramatic differences later? It seems intuitive (I know) that many systems which don’t fit the technical criteria of chaos still ought to be unpredictable in the way that weather is.
3) I liked Crichton’s early work. [I don’t like reading author’s misinterpretations of technical subjects I think I understand (but that’s a rather limited area).] I really disliked Crichton’s abuse of his popularity as an author to pretend to be a competent science advisor (eg his private meeting with the previous president).

53. valhar2000

I’m wondering what your sources are, and I’d love a reasonbable explanation as to why these conditions aren’t mentioned in most popularisations.

They are mentioned in Ian Stewart’s “Does God play dice”, and in “From here to Infinity”, and in other books by other authors whose names have escaped me. Still, not that hard to find.

54. JEB

Are there names for systems that have some of the three properties of chaos, but not all of them?
Of the three, only sensitivity to initial conditions strikes me as having obvious real-world consequences (like making weather unpredictable), so if a novelist were to focus on that property and ignore the others, what is the harm?
Also, I have always thought that the three body problem was an example of chaos, but maybe I’m wrong. Given that one of the bodies can escape the system, how can you have the property of dense periodic orbits? No matter how much time passes, the escaped body is never coming back!

55. fvngvs

I’ve been looking for a good, beginners guide to the math behind chaos theory ever since I did my bachelors thesis.
Roll on: the butterfly effect (it’s very pretty when you plot it), the logistic equation, self-similarity…
I may *finally* get to understand this lot with a bit of help, Mark!

56. Gilbert Bernstein

Mark,
A suggestion. Perhaps it would be a good idea to include large, well indicated disclaimers in posts like these. I think some of your readers might be confusing parts of this post with your normally lucid writing, leading to confusion and/or disappointment (e.g. comment #32).
Personally, I loved the post. I didn’t actually know what a chaotic system was until reading your post. But I think my joy here is contingent on already knowing what a phase space is and having studied Topology, so that “dense” is far less dense to me (Ba Dum Chh).

57. MJ

First, I apologise for drunkenly posting here above. Second…
“The phase space has the property that if the system passes through a point p in the neighborhood P, then after some finite period of time, the system will pass through another point in P. That’s not to say that it will repeat exactly: if it did, then you would have a repeating system, which would not be chaotic! But it will come arbitrarily close to repeating. And that almost-repetition has to have a specific property: the union of the set of all of those almost-cyclic paths must be equivalent to the entire phase-space itself. (We say, again speaking very loosely, that the set of almost-cycles is dense in the phase space.)”
This isn’t quite what ‘dense’ means. Dense used as above means the closure of the union-of-the-set-of-all-of-these-almost-cyclic-paths must be the phase space itself. In other words, if we pick any point in our phase space and look at any arbitrarily small surrounding neighbourhood, we’ll find points that are in our almost-cyclic path.
(So, as an example, imagine a path that passed through every point with rational co-ordinates, and only those points. This path would be dense in that any point would be ‘infinitely close’ to the path, but there wouldn’t be any points on our path with irrational co-ordinates.)
It’s possible you wanted to avoid this by using the word ‘equivalent’ in your penultimate sentence above; if so, I apologise for the nitpicking.

58. Dave

Re Heinlein: I recall reading that in his stories that involved interplanetary travel, he actually would compute the orbits involved. So he honestly did the math, when he could. And I think he did try to get his science right.
The real problem with a lot of science fiction is that the author wants to be able to have things that, as far as we know, aren’t possible in the real world, like faster-than-light travel. So they just declare it possible, and make it part of the story. Science goes out the window in those cases.
For some reason, in his last novels, Heinlein got a major bee in his bonnet about Cantorian transfinites. Why he had such enmity towards this concept, I have no idea. It’s one bit of mathematics I really did study in college, and I never understood where Heinlein was coming from on this one.

59. Pat Cahalan

I figured that the comment thread would be amusing, but this…
“Michael Crichton: The thinking man’s Ayn Rand.”
You sir, are an elevated thinker. That was *brilliant*.

If it had not been for Jurassic Park, I would have never ever learnt about Chaos Theory. Crichton was courageous enough to include graphs, diagrams and flowcharts in Jurassic Park and that alone earns my respect for him.

61. chris

@Paul
So in your world, as long as writer is entertaining, that writer is “good”? I don’t want to derail the Chaotic Crichton conversation, but it seems to me that that LOLcatz are now in line for the Pulitzer.
I would argue that when we’re talking about a writer (or whatever) “popular/successful/money-making” and “good” aren’t necessarily the same thing. Danielle Steele sells a lot of books… “good” writer? I think not. Her books are brain candy (at best).

62. John M 307

For example, I remember one of his where the main character is talking to a Wise Old Man Engineer in a spaceship, asking about why they couldn’t go faster than light. Heinlein knew that you couldn’t go faster than light, and he knew it had something to do with Einstein, so he had the engineer babble a bit – but he didn’t know anything about *why* you couldn’t, or what would happen as you accelerated. So he just made it up.

I don’t think Heinlein himself got it wrong in “Farmer in the Sky.” Rather, I think that he made the engineer not understand the issue properly. He always created imperfect characters in his heyday.
From the perspective of decades in the future, Heinlein got a number of things wrong in science, but I can only recall (right now) one thing where he was wrong himself from the perspective back then. I think it was in “Starman Jones” where he mentions that one planet had a moon, making it possible to measure its mass. The mistake was in forgetting that the orbit of the spaceship itself could be used to find the planet’s mass.

63. Robert Grumbine

Mark: I’m looking forward to the rest of the posts on this. Chaos is an old friend. If you’d like some citations to the early (1960s) Lorenz papers, drop me a line. (They ought to be easy to find, but once we’re that far back, I’m seeing a lot is actually hard to get to.)
Re SF writers and science: SF writers are writing fiction. Sometimes well, sometimes not. But fiction is the core. Some writers dress up their fiction with more excuses about the science (Verne, say, versus Wells’ inventing magical substances whole cloth (cavorite)). But it’s fiction.
I confess I’ve moved more towards that view of SF as I learned more science. It turned out that while SF tends to be more or less good with elementary physics (meaning physics of some easily counted number of discrete bodies), it does not do well with more complex physics (fluids, atmospheres, oceans, …) or biology.
On Heinlein specifically: He, like many, completely failed to understand tides. He references the tidal force as being inverse square (like gravity), rather than inverse cube (which it truly is, see, for example, A. E. Gill’s Atmosphere-Ocean Dynamics, or the much earlier book on tides by George Darwin). The Cat Who Walks Through Walls is the title which stayed in mind for this, but I’m reasonably confident that it shows up in his previous 40 years of writing as well.
But … it’s fiction, so once you suspend ‘correspondence to reality’ as a criterion, you can get on with ‘does the story make sense in its own terms’ and the like. I don’t think that was one of his best books, but it was a pleasant enough read.

64. Tim Gaede

I once took a programming course in chaos but we really didn’t go into depth about the subject. However, that was my first time writing anything graphical in a computer program and I found the experience entertaining.
Newton’s three-body problem is a bit interesting. I placed a planet in various habitable orbits around a star in the Alpha Centauri system. The planet’s angular momentum and total energy with respect to the star barely changed but the orbital characteristics (periapsis and apoapsis) varied substantially in a stable cyclical pattern with periods lasting a thousands of years.

65. Jonathan Vos Post

6-years obsolete, but shows why I agree with MCC:
[John] Michael Crichton (23 Oct 1942-) also known as John Lange and Jeffery Hudson:
one of America’s most profitable science fiction/mystery authors, film and television
major power. Formerly a doctor, he has a strange fascination with, and yet
profound distrust of, all technology, and has been criticized by Speaker of
the House Newt Gingrich for endlessly replaying the “Frankenstein” theme,
that there are some things not meant to be messed around with by Man,
which Gingrich contrasted with the optimistic technophiliac works of
Arthur C. Clarke.
Born in Chicago; A.B. summa cum laude 1964 Harvard University
(Phi Beta Kappa); M.D. 1969 Harvard University Medical School;
1969-70 Salk Institute, La Jolla, California; 1965 married Joan
Radam (divorced 1971); 1978 married Kathleen St.Johns (divorced
1980). Winner 1970 American Medical Writers Award; winner 1968
and 1980 Edgar Allan Poe Awards of Mystery Writers of America.
* Science Fiction novels include:
* The Andromeda Strain [New York: Knopf, 1969; London: Cape, 1969; Dell]
* The Terminal Man [Knopf, 1972; London: Cape, 1972; Bantam]
* Westworld [New York: Bantam, 1975]
* Eaters of the Dead [New York: Knopf, 1976; London: Cape; Bantam]
* Congo [New York: Knopf, 1980; London: Allen Lane, 1981]
* Jurassic Park {to be done}
* Mystery/Detective novels include:
* Odds-On [New York: New American Library, 1966] under pseudonym John Lange
* Scratch One [New York: New American Library, 1967] under pseudonym John Lange
* A Case of Need [Cleveland: World, 1968] under pseudonym Jeffery Hudson
* Easy Go [New York: New American Library, 1968] under pseudonym John Lange
also known as “The Last Tomb” by Michael Crichton [Bantam, 1974]
* The Venom Business [Cleveland: World, 1969] under pseudonym John Lange
* Zero Cool [New York: New American Library, 1969; London: Sphere, 1972]
under pseudonym John Lange
* Drug of Choice [New York: New American Library, 1970;
also known as “Overkill” London: Sphere, 1972]
under pseudonym John Lange
* Grave Descend [New York: New American Library, 1970]
* Dealing; or, The Berkely-to-Boston Forty-Brick Lost-Bag Blues
[New York: Knopf, 1971] under pseudonym Michael Douglas
(co-author Douglas Crichton)
* Binary [New York: Knopf, 1972] under pseudonym John Lange
* The Great Train Robbery [New York: Knopf, 1975; London: Cape, 1975]
* Screenplays:
* Westworld, 1973
* Coma, 1978
* The Great Train Robbery, 1978
* Looker, 1981
* Teleplays:
* Pursuit, 1972
* Miscellaneous Books:
* Five Patients: The Hospital Explained [New York: Knopf, 1970;
London: Cape, 1971]
* Jasper Johns [New York: Abrams, 1977] Artist biography/study
* Electronic Life: How to Think About Computers [New York: Knopf, 1983]
* Reference:
* Michael Crichton: A Critical Companion [Greenwood Press, 1996]
by Elizabeth A. Trembley
ISBN 0-313-29414-3, \$29.95, hardcover

66. Peter G

“But if you try to extend the predictions out in time, what you find is that they become unstable. Make a tiny, tiny change – alter the flow at one point by 1% – and suddenly, the prediction for the weather a week later is dramatically different.” How true. I find the faith people have in computer models of climate quite touching in the light of this fact and also that that problem is orders of magnitude more difficult and that the input data is dubious accuracy. Oh well “Computer says no.” That should be good enough. I will offer one correction in the interest of accuracy. The Navier-Stokes equations describe only fluid flow and by themselves can not be used to model weather with any accuracy at all there being many factors that influence weather that are not describable using only those equations.

67. Philip Travers

Read a book or two on Chaos theory,and the other night,Australian time,came across an article, about women in maths in a American newspaper on-line and she was into model theory.So somehow,I could look up my computer history,but wont,came across a Sierpinkski triangulation pattern.And noticed that this had inspired a solution in communication physical devices-gadgets.As someone who has been interested in pyramid power and even the Russians have groups working on this,and the effects on matters under pyramids…In a recent Nexus NewTimes Feb-March2009 Vol 16 No4Magazine article,Tesla Technology and Pyramid Power,are some suggestive matters involving the influence of Pyramids on weather.Chaos would seem to be a very diverse theory,and probably people like yourself, will find new uses and applications. Website http: //www.scribd.com/hrvojezujic. Too late in the night to be more forthcoming.

68. Alterin

Great post. Of course you knew that the moment you finished writing it but I guess you still like hearing that. π
I wonder how you could call other people fools as I have noticed that every time I call someone a fool I turn to be a fool myself.
Well, that was a meaningless post, but not the first one here. π

69. Hank Roberts

> tidal force
Good treatment here:
http://www.npl.washington.edu/av/altvw63.html
The Force of the Tide
John G. Cramer
Alternate View Column AV-63
January 1994 issue of Analog Science Fiction & Fact Magazine
“… Tidal forces result from imperfect cancellation of centrifugal and gravitational forces a distance L away from the center of gravity of the system and have the form Ft = G m M L / R3.
The tidal force is therefore an “inverse cube law” force.”