In the book and movie of “Jurassic Park” one of the main characters, Ian Malcolm is a Mathematician that specializes in Chaos Theory and non-linear equations. I have two questions. One, is Chaos Theory real? Two, what does a non-linear equation look like?
Chaos Theory is a real thing.
http://www.duke.edu/~mjd/chaos/chaos.html
Algebra, but not just ANY algebra–weeeeeeird algebra. 
http://www.autolev.com/WebSite/SampleProblemNonlinearSolve/NonlinearSolve.html
http://www.autolev.com/WebSite/SampleProblemVehicleSkid/VehicleSkid.html
Yes, Chaos Theory is real, all right. Crichton intended Jurassic Park as a sort of cautionary tale showing that chaos theory may indicate some huge dangers in the indiscrimante use of biotechnology.
I wouldn’t know a non-linear equation if it bit me in the ass, but a lot of the basic concepts can be understood without a math phd. One of them is “sensitivity to inintial conditions”, which is what Crichton was warning about. Basically, any uncertainty in an initial state, no matter how small, will lead to rapidly growing errors in efforts to predict future behavior. For example, a screensaver with a fractal pattern. Small perturabtions in the original pattern lead to erratically beautiful large scale patterns.
James Gleick’s book “Chaos” explains it much better than I ever could and is well worth the read.
God, my typing sucks.
God already knows. She just wants you to keep practicing at it.
You didn’t ask, but I’ll say it anyway; yes, chaos theory is real, but Dr. Malcolm is a complete and utter fake. Michael Crichton understands so poorly, and puts such ridiculous words in the man’s mouth, that real mathematicians must cringe.
The real paleontologists sure cringed. Gotta give Crichton that: he’s consistent.
Eh. It was a long time ago that I read Jurassic Park, but I don’t recall anything so horrible. On the other hand, I wasn’t expecting Crichton to get any details right, so maybe I was more inclined to be forgiving.
I’m not going to get all the terminology right, but this should give you a basic idea. I recommend a book called Chaos by James Gleick for further reading.
So anyway, let’s get onto this…
At bottom, chaos theory deals with iterated systems. An iterated system is one in which the behavior of the system is determined by what’s happened before. A very simple iterated system is given by the sequence <x[sub]n[/sub]>, where x[sub]0[/sub] is some real number, and x[sub]n+1[/sub] = 5/9 * (x[sub]n[/sub] - 32). You may recognize this formula; it’s the formula for converting fahrenheit temperatures to celsius. The interesting thing about this system is that no matter what you choose for x[sub]0[/sub], the sequence you get will eventually converge to -40. This is a very simple linear system.
What’s a linear system? Well, suppose your variables are x, y, and z. A linear system is one where new values of x, y, and z are determined by taking the old values, multiplying them by a constant, adding these together, and possibly adding another constant to each one. Here’s an example with two variables: x[sub]n+1[/sub] = 1/6 * x[sub]n[/sub] + 1/4 * y[sub]n[/sub] + 1/2, y[sub]n+1[/sub] = 1/3 * x[sub]n[/sub] + 1/5 * y[sub]n[/sub] + 1/8. This system also converges to some fixed pair (x, y), and we can discover that by setting x[sub]n+1[/sub] = x[sub]n[/sub] and y[sub]n+1[/sub] = y[sub]n[/sub], and then solving. This is left as an exercise to the reader.
A non-linear system is pretty much anything else. If you square the variables, multiply them, take their quotient, etc., etc. then it’s a non-linear system. These may also have the convergence property I mentioned above, but they can have other properties as well, some of which may be very strange.
The matter of interest for practical purposes is error propagation. Suppose I take the first system I gave you, and plug in two initial values, x and x + [symbol]e[/symbol]. What’s the difference between x[sub]1[/sub] in the two cases? Well, it actually turns out to be 5/9 * [symbol]e[/symbol]. In fact, the difference between x[sub]n[/sub] in each case will be (5/9)[sup]n[/sup] * [symbol]e[/symbol]. As n gets bigger and bigger, the error gets closer to zero, so that’s why you get convergence.
The important point is that in a linear system, the error obtained after a finite number of iterations is a constant multiple of the original error. If the original error is very small, it’ll take a long time before you notice any differences in the two systems (if you do at all).
Non-linear systems aren’t like that. Error can propagate very quickly (relative to the fact that we’re allowed any finite number of iterations, at least in theory). Consider the error propagation in x[sub]n+1[/sub] = x[sub]n[/sub][sup]2[/sup], x[sub]0[/sub] = k, a real number. If the error after n iterations is [symbol]e[/symbol], the error after n + 1 iterations is 2[symbol]e[/symbol]x[sub]n[/sub] + [symbol]e[/symbol][sup]2[/sup], which could be very large (try a few iterations for x[sub]0[/sub] = 10 and [symbol]e[/symbol] = .1).
Now for the important question: who cares? Well, basically anyone who’s trying to model the real world. Our best models for most parts of nature are non-linear systems, and we’d like to use them for predictions. This generally requires calculations done by computers. Now the thing that sucks about computers is that they can’t store values to infinite precision; they have to round off (in fact, .1 can’t be stored exactly by a computer, cause it’s a repeating “decimal” in base two). Know what that means? Error, and lots of it! The error introduced by rounding off is propagated through the system, and after potentially very few iterations, the predicted behavior of the system is very different from what actually happens. That’s why your weatherman can’t get his predictions right more than two days in advance.
So what about other strange properties of non-linear systems? Well, one might think that similar initial conditions (the input you give at first) would lead to similar outcomes, but that ain’t so. In fact, for some systems, there are inputs which lead to stable behavior (think convergence like in my first example), and other inputs which lead to very unstable behavior (like oscillation between large positive and small negative values), and the difference between these inputs may be smaller than rounding error. Sorry, I don’t have any examples, but a web search might turn up something (up for it, DDG?).
I hope this made sense. Check the book I referenced earlier for more detail, and feel free to ask questions (but I don’t promise to be able to answer them!).
Can anyone explain the math behind this:
Given: The basic principles of chaos theory
Prove: Cloned dinosaurs WILL run amok, no matter what precautions you take, and they WILL bite you in half if you hide in an outhouse?
I’m having trouble with some of the intermediate steps.
Thanks in advance.
[sub]I promise that it’s not for a homework assignment.[/sub]
Only Michael Crichton’s half-assed “science” can explain that. There’s no support for it in the real world.
I never bought the chaos theory arguments in the novel. A friend of mine pointed out that the mathematician saying “you can’t do this because mathematics proves it will go wrong!” was just a stand-in for the religious-type who would be saying “you can’t do this because there are things man was not meant to know!”
I never understood why breeding dinosaurs was such a big danger. They’re all stuck on an island, and even if they somehow got to the mainland it’s not like they could threaten humanity. All you’d have to do is announce “Hey, hunters! There are big dangerous raptors loose in South America!”
Within days, the place would be crawling with hunters carrying nightscopes and explosive rounds. Even if half of them got killed in the process (which would make it even more tempting for thrill-seekers), they could easily re-extinct the dinos.
Hey! Go easy on Crichton! Andromeda Strain was an excellent book! Sure some of his science is crap (especially Time Line), but that’s why it’s fiction, no?
I have never understood Crichton’s chaos-theory-means-dinosaurs-will-eat-the-guests connection; Jurassic Park was basically a big zoo, and we’ve been running zoos for how many generations? If we can keep lions and elephants under control, why not dinosaurs?
I had this argument with my best friend at the time, who dogmatically insisted that because chaos theory shows you can’t predict everything, a real life Jurassic Park would indeed fall apart. He couldn’t explain why a zoo works. For that matter, I tried to get him to explain how a CITY works. A city’s more complicated than a zoo; why aren’t all cities in rubble?
BUT: It was a fun book and a fun movie, so who’s complaining?
That’s true of chaotic systems, but it’s not necessarily true. Looking back through the posts, it’s easy to see that that is actually the bone of contention–is the specific system chaotic or not? Are there variables that cause flucuations in the system to damp out?
It’s hard to prove that a system is chaotic, in real life. But it’s also good to recognize the possibility.
A real-life Jurassic Park would indeed fall apart if it was as poorly designed as the one in the movie.
Lets start with the basics. How many zoos have you been to? How many of these zoos used electric fences to keep the animals in their pens? And, how many of those zoos used their electric fences as the only means of keeping the animals under control?
Then, if you decide to use nothing but electric fences, wouldn’t you design the system so that the standby generator turns on automatically when the main power fails?
That’s just one example. I’ll also point out the problems in JP had more to do with deliberate sabotage than it did with with chaos theory.
The problem I have with most of Crichton’s books is that much of the time the danger the characters are in is caused by a glaring oversight. Yeah, people miss things, but usually not as obviously as the characters in his books.
Jurassic Park - “We’ll use electric fences to keep the dinosaurs in check!”
“But, what if the power fails?”
“Don’t worry, we’ll send someone across the Park on foot to start the emergency generator.”
Andromeda Strain - “We’ll have a nuclear device destroy the base if the virus escapes.”
“But, there are five sealed sections and there are only overrides for the devices in three of them.”
“Don’t worry, someone will run to the other sector before it is sealed off if it is a false alarm.”
Terminal Man - “We’ll set up our device to stimulate his pleasure centers whenever he goes into a psychotic rage.”
“But, wasn’t there something in the Psychology books about positive reinforcement?”
“Don’t worry, if he goes out of control we’ll… uh… shoot him! Just watch out for the reactor we implanted in his body.”
You get the idea…
Not to say Crichton isn’t an entertaining writer. It’s just that some of his set-ups make me cringe when I read them.
You really liked that?
‘AHHHHHHHHH!!! A SPACE VIRUS WILL KILL US ALL!’
‘It’s escaped and it’s going to eat Los Angeles!’
‘AHHHHHHHH!!!’
‘Oh wait. It mutated. Now it is harmless. Let us have a lollipop.’
Sorry. I just thought the ending was weak.
You know, it would be worth cloning dinosaurs and creating a vast, hubris-laden, triumph-of-man style park, just to demonstrate that Chaos theory doesn’t prove squat, in this instance.
<SPOILER ALERT (as if anyone hasn’t seen the movie or read the book)>
I mean, look at the movie; the park collapsed because of one silly, obvious flaw: too much power in the hands of one man. Who? Nedry. He was known to be untrustworthy, he had some sort of unnamed beef with what’s-his-name, the park creator. He designed all the computer stuff himself, and when he tried to steal the embryos, he alone was responsible for the park power shutting down. Even half-assed IT-related companies have multiple guys who build and know the infrastructure…you’re telling me that this huge, visionary project would employ ONE FREAKIN’ GUY to do everything that kept the park running?
I ain’t buyin’ it.
Had they not hired Nedry, why would the park have collapsed?
Crichton’s science has always been poor. Despite the fact that he has a medical degree, it appears that he has no idea of (and no interest in learning about) how to do research in the subjects about which he writes. The scientific details in his books and movies sound like he read a newspaper article about the subject five years before and didn’t bother to read anything else about it.
Furthermore, a consistent theme in Crichton’s works is that “there are things man was not meant to know (or to do).” Notice that both the Jurassic Park and the Westworld films are about amusement parks gone wrong. Is his conclusion that all large amusement parks are evil? Great, let’s close down Disneyland and Disneyworld.
The logic is just as simple as:
- Collect underpants.
- Profit.
Well, animals escape from Zoo enclosures all the time. The San Diego Zoo had a Bornean Orangutan named Ken Allen that got out nine times over his life. He was once observed teaching another Orangutan how to use a stick as a tool to escape.
A google search will turn up a zillion hits on fun “Escape to the Zoo” promotions, and a smattering of hits on actual animal escapes. And these are the ones that made the papers…
Maybe a better (though much less sexy) systems-oriented take on the inevitable failure of Zoo enclosures would have been John Gall’s work “Systematics: How Systems Work and Especially How They Fail.” Some of his theories are:
*Systems in general work poorly or not at all.
*New systems mean new problems.
*Complex systems usually operate in failure mode.
*When a fail-safe system fails, it fails by failing to fail safe.
The last one parses awkwardly, but is self-evident if you think about it. Take, for example, an elevator fail-safe system designed to mechanically arrest the fall of the car in case of cable and/or brake failure. The system is designed to prevent the car from entering an uncontrolled descent. If it engages when not needed, it has still not allowed the car to fall, so the system has not failed, it has had a malfunction. There is only one way for the system to fail, and the results are not pleasant (unless, of course, you happen to live in a world with cartoon physics, where all you have to do is jump up just before the car finds the bottom of the shaft). It is also rather difficult to comprehensively test a fail-safe system without intentionally introducing the very risk the system was designed to obviate in the first place.
Since I started barely on topic, and have now wandered completely off, it is a good time to stop 8-).
I have the misfortune to work with some very large systems, and I find John Gall to be a hoot.
–jack