A Short Introduction to the Jargon of Iteration Theory ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ Iteration theory (just as every other complicated field) has developed its own jargon. This list includes some of the more common terms. It may help you understand some of the other documentation better, and it may help you understand iteration better as well. And if all else fails, you can use these spiffy mathematical terms to impress your friends with your vast stores of chaotic knowledge. Dynamical System ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ A dynamical system is simply a function together with the domain the function is defined on. The domain can be anything--a line, a line segment, the plane, 3-space, 6 dimensional space, or any of the other weird "spaces" mathematicans are always coming up with. (In Iterate!, the domain of the function is always the plane.) The only restriction is that the domain and the range of the function must be the same. Symbolically, we would write: f: D  D This means that 'f' is a function with domain and range D. This requirement makes sense if you think about it. When you iterate a function, you keep feeding points from the range back into the domain. So if the range and the domain aren't the same, you're going to be in trouble. The reason this is called a "dynamical system" is that "dynamics" means "movement". What we are studying when we look at a dynamical system is how the points move around under the influence of the function. Iteration ÄÄÄÄÄÄÄÄÄ What we do when study a Dynamical System is "iterate" the points. This means you start with a point x. Then figure out f(x). Then f(f(x)), f(f(f(x))), f(f(f(f(x)))) and so on. Writing all this f(f(f(f(f(x))))) stuff gets pretty tiresome, so mathematicians abbreviate by writing fü(x). This means that you apply function 'f' to point 'x' 'n' times. So fý(x)=f(f(x)) and so on. (It would be easy to get confused and think that fý(x) means "f(x) squared". To distinguish between the two, mathematicians write (f(x))ý if they mean "f(x) squared." It would also be easy to get confused and think that fý(x) means "the 2nd derivative of the function f." But if you're smart enough to take the second derivative of the function f, then you should be smart enough to tell the difference between fý(x) meaning "the second iteration of f applied to x" and fý(x) meaning "the second derivative of the function f.") Orbits ÄÄÄÄÄÄ What you are interested in looking at in a dynamical system is the path the points take when they are iterated. This path is called the "orbit". Another way of saying the same thing: The orbit of point x consists of these points x, f(x), fý(x)), . . . , fü(x), . . . The orbit of a point is what you see in Iterate! when you press . Fixed points ÄÄÄÄÄÄÄÄÄÄÄÄ Fixed points are points that don't go anywhere when they're iterated, that is, x=f(x)=fý(x) etc. Another way of saying the same thing: The orbit of a fixed point consists only of the point itself. Periodic Points ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Periodic points are points that come back to the original point after a certain number of iterations. For instance, a period 2 point comes back to the original point after two iterations: x (starting point) f(x) (a different point) fý(x)=x (back to the starting point) Periodic points of every different period are possible. Once a periodic point returns to the starting point, it just repeats the same points again until it reaches the starting point again. For instance, here is a possible orbit for a period 5 point: 0, «, 1, 1«, 2, 0, «, 1, 1«, 2, 0, «, 1, 1«, 2, 0, «, 1, 1«, 2, . . . As you can see, it just keeps repeating the same 5 points over and over. So the orbit of a period 'n' point consists of just 'n' points. Attracting Orbits ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Attracting orbits suck nearby orbits closer and closer to them. For instance, an attracting fixed point sucks all nearby points into itself. A period 3 attracting point sucks all points near its orbit closer and closer to the orbit (the orbit consists of three points, of course). Repelling Orbits ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ A repelling orbit drives nearby orbits away from it. Other Types of Orbits ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Many other types of orbits are possible. For instance, there are fixed points that are attracting in one direction and repelling in the other. By using techniques from elementary calculus, it is relatively easy to tell which orbits will be attracting, repelling, or something else. Check the literature for more details on this. Using Iterate!, you can easily find examples of all of these different types of orbits (fixed points, periodic points, repelling orbits, attracting orbits, etc.). You may have to try several different functions with different parameters, and try iterating several different points in different areas of the plane for each of them, but eventually you will see all these different types of orbits. Strange Attractors ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ A strange attractor is similar to an attracting orbit. The difference is that in an attracting orbit, everything is attracted into an orbit which consists of a finite number of points. We would say the it is a "finite attractor". A strange attractor, however, is an "infinite attractor". That is, there is an infinite set of points that everything else is attracted to. Where the attracting orbit consisted of only a few attracting points, you can think of a strange attractor as being a whole shape that is attracting. Usually this shape is a very, very weird shape; that is why it is called a strange attractor. As a rule, the strange attractor is a fractal, with fractal dimension less than dimension of the dynamical system. For instance, in Iterate!, we are iterating functions on the plane, which has dimension 2. So any strange attractors we find in Iterate! will have dimension less than 2--say 1.7, 1.2, or 0.5. Usually, the dynamical system is chaotic on the strange attractor. It isn't chaotic on the rest of the dynamical system, though, since the rest of the system is just sucked up into the strange attractor. (See below for the definition of chaos.) To see a good example of a strange attractor, select the Horseshoe Map (Function L) with default window and parameters. The "Horseshoe" shape that you see when you iterate a point (which actually consists of horseshoes within horseshoes within horseshoes) is a strange attractor. You will notice that all points are drawn into this horseshoe shape--it is an attractor. You will notice that once a point gets close to the horseshoe shape, it seems to just jump around randomly on it--it moves chaotically on the strange attractor. The horseshoe shape appears to have a fractal dimension between 1 and 2--probably about 1.4 or 1.5. Another example of a strange attractor is Function F (the inverse Julia Set function). Again, the strange attractor is a fractal with fractal dimension between 1 and 2. Although strange attractors _are_ strange (hence the name), a dynamical system with a strange attractor is often easier to understand and analyze than one without a strange attractor. Forward and Reverse Orbits ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ To make the reverse orbit of a point, think of running the function backwards. In other words, instead of applying the function to the point repeatedly, you apply the inverse function of to the point repeatedly. All the points you get by doing this are the "reverse orbit". Another way of saying the same thing: The reverse orbit of a point 'x' is all the points that are mapped to 'x' under iteration. In other words, if fü(y)=x, then y is in the reverse orbit of x. If mathematicians are talking about "reverse orbits", they will often refer to the normal orbit as the "forward orbit" just to be clear. If they are talking about "forward" and "reverse" orbits, then usually just plain "orbit" means the forward and the reverse orbits together. (Hey now, let's not hear any complaints about this--you don't expect clarity and consistency from a bunch of mere mathematicians, do you?) In Iterate!, Function F is the inverse of Function E. So if you iterate a point under Function E, you get the forward orbit of the point. If you iterate the same point under Function F, you get the reverse orbit of the point. Chaos ÄÄÄÄÄ Mathematically, chaos is defined as a dynamical system with certain (chaotic) properties. In your own personal life, you are welcome to define chaos any way you want (most of us don't need to define it actually--we just live it). But you might want to know the "official" definition of chaos as well. So here it is: A chaotic dynamical system must satisfy three properties: 1. Sensitive dependence on initial conditions. This means that any two points that are close to each other must end up far away from each other after a few iterations. This condition ensures that the points are thoroughly scrambled up. 2. Topological Transitivity. This is a more technical requirement, so I won't try to explain it. Basically, it insures that every area of the dynamical system is scrambled--there aren't some small pockets somewhere that don't become scrambled. (See "An Introduction to Chaotic Dynamical Systems" if you want more info on this.) 3. Periodic points are everywhere dense. No, this doesn't mean that all periodic points are stupid. It just means that any region in the dynamical system--no matter how small--contains a periodic point. You can think of a chaotic dynamical system as one that is thoroughly mixed, and scrambled; the points move as though at random; the movement appears to be unpredictable. If you like homey analogies, you can think of a dynamical system as being like mixing bread dough. A chaotic dynamical system is like thoroughly mixed bread dough; a non-chaotic dynamical system is like dough that isn't well mixed. If Properties 1, 2, and 3 happen in the mixing of the bread, then we can be sure that it is well mixed: Property 1 ensures that things that started out close together end up far apart. For instance, the flour that we put in all together at the start isn't still clumped up all together--it's spread far and wide. Property 2 ensures that everything is mixed throughout the _entire_ dough. For instance, the oil we put in isn't just mixed around in one little corner of the loaf, but is evenly mixed throughout ALL of the dough. Property 3 assures us that although the mixing process seems to be "chaotic", disorderly, and generally difficult to understand, behind this chaos is a very strong order, dependability, and even simplicity (remember that the periodic points are about the simplest kind of motion we can have, and Property 3 assures that they are scattered throughout our bread dough). (*see note) In the case of bread-making, this order, dependability, and simplicity is best understood as a result of the result of the kneading process. Kneading is very simple--a couple of simple motions are repeated over and over in a sort of "iteration" of motion. And although it is "chaotic", it is dependable and reproducible, too--every time we knead bread dough, we end up with the same basic result. *Note: Although everywhere dense periodic points are an important feature of the mathematical formulation of chaos, there is a valid question about whether they would actually appear in a physical representation of a dynamical system, i.e., in bread dough. A mathematician would instantly anwser, "Yes, of course they do! Or at least something so close to periodic points that you couldn't tell the difference." A physicist might say, "Due to the fact that space and time are ultimately discrete (in the 32nd dimension--but let's not get into that), and after all, there are only a finite number of elementary particles in the universe, let alone in a blob of bread dough, ALL the points in the dough are ipso facto periodic and there's NO SUCH THING as chaos in bread dough or real life." (The physicist could easily be disproven by a brief tour of my apartment.) A really sane person might come up with yet another answer. In any case, the question is a good one, and not easy to answer. Map ÄÄÄ "Map" is simply another word for "function". The two words mean exactly the same thing. For some reason, iteration theorists often use the word "map" instead of "function". What Good Is It? ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Usually when mathematicians are asked this question about their specialty, they answer, "It expands the realm of human knowledge," "It challenges our intellect," "In about 12,000 years it might be able to be applied to some obscure scientific area," and stuff like this. With Iteration Theory, though, we don't have to get into these flimsy type of justifications (just as though someone ought to be paid just for thinking... hmmph, the gall of those mathematicians). Iteration Theory has a ton of concrete physical applications. One obvious application is modelling population growth. Biologists typically think of population change on a yearly basis. The trick is to find an equation that will tell you next year's population if you know this year's. (If you read "Function.txt" you will see that several of the functions that are programmed into Iterate! were made with this kind of biological idea in mind.) So if we have such a function, and we know this year's population, we just apply the function and Presto! we have next year's population. Apply the function again and we have the population in two years. Apply it again, and we have the population in three years, and so on. And what is this? A Dynamical System, of course. In fact, I have a book on my desk right now called "Chaos and Insect Ecology." The authors talk about such things as whether the conclusions of Chaos Theory can be applied to insect population dynamics; they apply chaos theory to things as diverse as the spread of measles in New York City and the population of martens in Canada. Most anything that moves or changes (and that includes pretty much everything) can be thought of as a Dynamical System and studied using Iteration Theory. Weather prediction, for instance, has been extensively studied from this angle. The most profound result of this study is the conclusion that the equations governing the weather are chaotic. This makes long term weather prediction impossible. The studies have shown that a change in the weather conditions as small as a butterfly's wing flapping can change the entire global weather pattern three months later. So unless you can account for every butterfly's wings, each person walking down the street, and other such changes that minutely affect the weather, you can't predict the weather more than three months down the road. (This "Butterfly Effect" can be observed in any of the chaotic functions in Iterate!. For instance, select Function I with default windows and parameters. Iterate a point with . Use to move to the very next point on the screen. You will see that the orbits of the two points aren't close to each other at all. You can use the

command at the command screen to enter points that are even closer to each other; then use to iterate them and to examine their endpoints. You will find that the endpoints aren't anywhere close to each other. This is the Butterfly Effect: a small change in the initial conditions leads to a large change in the end result. This also the basic idea behind "Sensitive Dependence on Initial Conditions" mentioned above.) The whole area of chaos theory-iteration theory-dynamical systems-fractals and so on is really a brand new field. Most of the major discoveries in Iteration Theory have been made in the 1980s. Although it is new, its impact has already been major. These new methods promise to transform the way we think about science and mathematics. With Iterate!, you can see for yourself many of these exciting discoveries, and maybe along the way you'll make a few of your own! (Ver. 3.11, 12/93)