For a dynamical system to be classified as chaotic, most scientists will agree that it must have the following properties:
- it must be sensitive to initial conditions,
- it must be topologically mixing, and
- its periodic orbits must be dense.
Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.
Sensitivity to initial conditions is popularly known as the "butterfly effect", suggesting that the flapping of a butterfly's wings over Tokyo might create tiny changes in the atmosphere, which could over time cause a tornado to occur over Texas. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.
Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behavior, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.
Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimension torus described by a pair of angles (x,y), each ranging between zero and 2π. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.
Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.
I don't even care about the second two points, because, much like many mathematical theories, the first point of chaos theory falls apart when applied to real life, specifically nature. There are patterns everywhere in nature, and yes, they may seem chaotic to us, but they usually have some sort of symmetry and balance and an underlying system. Here are some examples, using some of the famous "butterfly effect" examples:A butterfly flaps its wings in Tokyo and a tornado happens in Texas, eventually. Why? Because the butterfly changing the miniscule amount of air around it causes the weather patterns to change, therefore creating a massively energetic meteorological phenomenon half a world away. Bull. Crap. That butterfly's wing flap injects a tiny amount of force and energy on to the surrounding environment. This energy is diffused, dampened, and scattered into relative nothingness within a few seconds by my friend and yours, entropy. The butterfly flapping its wings doesn't matter because compared to the massive amounts of energy coming in from other sources, namely the sun and moon and earth's core, the energy expelled by Mr. butterfly is nothing. Negligible. Irrelevant. In the big scheme of things, it changes nothing.
Perhaps a more concrete example is dropping a pebble in a pond, another popular chaos theory "lets bring it to the masses" example. Again, this is in nature, and again, it doesn't hold up. Dropping a pebble in a pond causes ripples. This is true. The ripples extend out radially. Also true. But if you stood on the other side of the pond, you could not detect that a pebble has been dropped way across the pond, now could you? No. Once again, its is because the energy diffuses, takes other forms, bounces off of a bank and is absorbed as impact, and so on and so forth. In fact, I'm looking out at sailboats and motorboats that are leaving wakes, sizeable wakes, in the harbor, and those waves are not even making an impact on the water close to me. Yes, the water close to me is exhibiting signs of disturbance, but much more from the big tankers and wind than any 20 foot sailboat cruising along. Is it detectable? Maybe, but that 20 ft sailboat, for the sake of predictions, can easily be ignored. Which brings me to my next point, size comparisons and total sample.
The previous two examples were both examples of a very small effect on a very large system. The small effect can usually be ignored in a large system. That brings us to the example used in Jurassic Park. That system was one drop, falling down a hand that may be what, 4 or 5 inches(10 cm) long, max? The changes in environment on the hand, namely veins pulsing, hand twitching, are miniscule to us, but not to the environment they are considered in. To the whole environment(the hand) they are what, 1:1000? (assuming a 1/10 mm twitch or pulse) And to the actual moving object, the drop of water, we're talking 1:5, maybe 1:10. If you dropped a 1 foot wide rock in a 1000 foot wide pond, you'll be able to detect it across the pond.
Or, in the butterfly case, you'd need a Mothra the size of Virginia to make a difference. An average hurricane delivers 10^14, or 100000000000000 Watts of energy per day, or 200 times the total energy production capability of mankind. To say this can be changed by something as small as a butterfly flapping its wings half a world away is ludicrous. This all brings me to what I think is the real failing of the butterfly effect, which is man's general self centeredness. Man tends to like to think he makes a difference in his surrounding world, and in a sense, this is true. Man can have an effect on his neighborhood, his town, possibly his state, and maybe even his country. Powerful people can affect the world of man. However, even the most powerful person in the world can not truly affect the Earth, let alone the solar system or galaxy. Even if every nuclear bomb was set off simultaneously, the earth, while devastated, would recover eventually, given enough time. Would it change? Yes, but in the grand astronomical scheme of things, nothing would change. Earth would continue to be a mostly water covered planet 93 million miles from its parent star. And it would still be able to sustain life. Meager at first, but eventually the earth would return to human-habitable status.
The butterfly effect supports this theory of "even the littlest change can have an effect". In reality, the littlest change often means nothing, and gets swallowed up and forgotten by the larger, more important changes.
How's that for a cheery outlook?
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