In the far distance, a Tyrannosaurus Rex roars and many Apatosaurs trumpet ... Chaos reigns Isla Nublar and Nature finally finds its way. Ian Malcolm has been right all along: the Jurassic Park is a nonlinear dynamic system, which is intrinsically unpredictable and unstable, and its safety is totally at the mercy of a handful of computers. A little greediness by the computer programmer Nedry (Nerdy?) has sent the whole park to its doom. Now the mighty and efficient flesh-crunching machines are set free, and in their eyes, their creators are but bunches of proteins and nutrients ...
Did everybody see Steven Spielberg's mega-hit movie Jurassic Park? You didn't? Well, you should. Oh, you did? Good boy. Did you like the dinosaurs? Good, Steven is awfully good at making those huge monsters and giving people a few delightful scares, and he always makes big bucks out of it.
Do you remember Ian Malcolm in the movie? No? Oh, it is that rock-star-like guy with black shirt, black trousers, black socks, black sneakers, black sun-glasses, black hairs ... black everything. Oh, not an African American, he's white.
Remember him now? Good. What did he keep on murmuring about in the helicopter when they were flying to Isla Nublar, the small island off the coast of Costa Rica, where the Jurassic Park was located? You remember? Good. "Chaos," yes! "Nonlinear dynamics," yes!! "Strange attractors," yes!!! "Butterfly Effect," yes!!!! Good memory, I wish I'd been that good. Don't worry if you could not understand any of those words. They aren't supposed to be understood anyway.
Image (right): Fig. 1. Jurassic Park.
Can anybody tell Malcolm's intention when he held Ellie's hand and put a few drops of water on her wrist, a little bit after their doomed Jurassic tour had started? Anybody? What did you say? He was flirting with her?? You nasty little boy! Stop giggling, you good-for-nothing bunch. He was trying to explain to her a complex system's sensitive dependence on initial conditions! What's a complex system? Hmmmm ... a girl is a complex system. Got it?! Good, you'll know as you get older.
Want to know something about the Butterfly Effect? Oh no, not Madam Butterfly. I know your parents brought you to the opera. This butterfly is not Japanese. This time it happens to be made in China. It goes like this: a butterfly flaps its wings in Beijing, and the stock market on Wall Street flounders. Don't think so? OK, let me modify it: a butterfly flaps its missiles, oops, wings, in Beijing, and the stock markets in Taiwan and Hong Kong flounder. OK? That's more like it. You know, Nedry to Jurassic Park is like the Butterfly to Taiwan and Hong Kong.
Image (above): Fig. 2. A nonlinear dynamic system. (Cartoon (c) 1988 by Jacques Boivin).
Who is Ian Malcolm then? Oh, he's one of the leading new-wave Chaos mathematicians who emerged from obscurity and had quickly risen into stardom in the '80s, riding the tide of the explosive growth of computer techniques. They, unlike traditional mathematicians who despise any connection of their work to the secular world, are openly interested in "how the real world works." Despite what the almighty and omnipotent Albert Einstein said about mathematics, "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." These Ian Malcolms have managed to do well. See, they are even portrayed in one of Hollywood's big movies, a treat few physicists, even the very Albert Einstein, have. Maybe they are quite certain that their mathematics is uncertain enough to be relevant to the real world.
These new-wave mathematicians work almost exclusively with nonlinear equations, either just for the fun of it or using them as toy models for real-world complex phenomena like the stock market, the weather, and ... the Jurassic Park. They also heavily use computers -- a practice REAL mathematicians frown on -- and computer graphics to generate dizzying and often intriguing colorful pictures which some REAL artists don't regard as art.
Yes, their mathematics may be uncertain; their usage and sometimes reliance on computers may be frowned upon; the computer images generated from their work may not regarded as art. But, they introduced a new way to look at how the real world works and how Mother Nature is organized and structured. They induced a paradigm shift. From their work, the word "Chaos" has gained new meanings and lost some of its mysteries.
Image (right): Fig. 3. Create chaos from order using simple rules.
(Cartoon (c) 1988 by Jacques Boivin).
Before the new theories of Chaos were developed, "chaos" had not been an auspicious word, and its ominousness was deeply rooted in many religions and cultures. Even today, when people speak this word, most of the time they mean disorder, confusion, something incomprehensible, something indescribable, something out of control, something bad, something that should be avoided at any cost. Although Chaos is also referred as the state of the world before Genesis, and that state may not be necessarily bad, it is still for sure that nobody would ever want to live there. "Chaos" had been used like the trash bin on a MacIntosh computer -- one can dispose into it anything that doesn't fit in with one's frame of mind and thus cannot be explained; then a purge takes care of it all.
Yet Chaos remains. You know it when you are caught by an unforecast rain storm, on a country highway, in a pitch-black night; you know it when the stock market dips suddenly and some broke stock brokers jump over the edge of a skyscraper; you know it when three years ago your skills were at the top of your profession and now you find them to be out of date and that you are out of job. How much human beings want to know the future and control their own fate! What should one do?
Some people resort to the almighty God; some people go to see crystal balls; some people hide their heads in the sand. But scientists, as always, call upon their intelligence and scientific methods. Scientists may oversimplify; they may just get a small glimpse of a tremendous natural force; they may only approximate; and along the path of search for truth and understanding, they may fail, but at least they have tried. Had God really existed, he must have favored scientists the most, since they bother him the least. God put intelligence in every human being he created and he wants people to use it by themselves when facing fate, so that he can have more time to drink. What do you think can make God angrier than interrupting his feast when he gave you the very intelligence not to bother him?
Now let's first think like a scientist and ask a few questions. Is Chaos really orderless? Could there be a hidden order in Chaos that regular methods fail to unveil? If we can not tell where a system goes next, can we tell where it might go? Does Chaos have to be in a complex system? Could it be that when we see only disorder and unpredictability, we might be too localized and narrow-sighted, so when we go up one or few levels higher, we might be able to see order and the whole picture?
Chaos theory's answers: No, Yes, Yes, No, Yes.
Image (right): Fig. 4. Chaos -- Orderly disorder.
Think about a hurricane. When you are aboard a ship amid a hurricane, all you feel is Chaos, even disaster. That's because you are too much inside it. If you happen to be in a weather satellite above the hurricane, then what you see from there is an orderly and majestic swirl moving along a certain path. Different view points sometimes do give qualitatively different understandings.
Chaos theories treat real world complex systems as nonlinear dynamic systems. A nonlinear dynamic system (see Fig. 2) is a system which evolves through time according to certain rules and which, when disturbed by external forces, doesn't respond proportionally with the degree of perturbation. In contrast, a linear dynamic system responds to perturbation linearly or proportionally. Take some real-life examples. A new car is a linear dynamic system; when you depress the accelerator pedal harder, the car runs faster; when you don't depress the pedal, well, it stops. A well-worn grandpa car, on the contrary, is a good old example of a nonlinear dynamic system; when you depress the pedal gently, it may jump like an excited horse; when you depress the pedal harder, it may hardly move; sometimes when you do nothing or very little things, it may tremble or simply disintegrate.
Chaos theorists use nonlinear differential equations to model the rules of nonlinear dynamic systems. Sometimes there can be just one simple rule; sometimes there are many complex rules. Simple rules do not automatically give rise to simple system behaviors (see Fig. 4). What a nonlinear dynamic system does is to follow the same set of rule(s) over and over again through time and space. The present state of the system is determined by the last state and in turn decides the next state, and so on. One of these spiral cycles is called an iteration.
If one simply follows one iteration after another too closely, he may get lost easily, just like being trapped in a hurricane. It requires a leap into a global view to see the whole picture, which is the result of all the iterations. Chaos theorists don't care about one particular iteration and where it will lead to; they care about the global picture -- the phase portrait. Fig. 4 is an example of phase portraits. In Fig. 4, if you plunge too much into the details, you may be like a mountain traveler trapped in a morning fog and don't know where you are and where you are going. Jump out and jump high; you will see an unspeakable order before your eyes!
MW Home Page
April Issue ToC