Sensitive dependence on initial conditions

Sensitive dependence on initial conditions (or SDIC as my lazy fingers prefer) is an important and often misunderstood concept that is relevant to both chaos theory and dynamical systems. In Golem's well crafted writeup regarding chaos theory, the author ascribes properties to SDIC which are really more the purview of a feature known as topological transitivity. This is a very common misunderstanding (or oversimplification). I will first provide a technical definition for SDIC and will limit the discussion to the domain of real numbers.


Definition
A function f which maps a set V onto itself has sensitive dependence on initial conditions if there exists d > 0 such that, for any x which is an element of V and any neighborhood N of x, there exists y (which is a member of N) and n >= 0 such that |f ⁿ(x) - f ⁿ(y)| > d.

Now, some of you may be asking yourself, what the hell does all that mean? More or less, it means that if you start off with a number x there is a number y that can be arbitrarily close to x, such that we could iterate the function f some number of times for both values and have the difference (|f ⁿ(x) - f ⁿ(y)|) be arbitrarily large. Note that this does not mean that all values in the neighborhood of x will behave in this manner, just that there is at least one.

So, in other words, even if you measure something with an arbitrarily small but non-zero error, you have the possibility of having a value that could diverge scads from what the "real" value would have done.

Using f(x)=x ³ as an example, we see a function can exhibit SDIC without exhibiting the wonky behavior that characterizes a chaotic dynamical system.

Throw in a dash of topological transitivity and that is when you start getting crazy with the cheese whiz. We have a property ( SDIC ) which insures that you will move arbitrarily far from at least one person in your neighborhood (or they will move away from you, depending on who was using the cheese whiz). In topological transitivity we will find a property that insures that the people in your neighborhood will have, over time, moved through every other neighborhood.


Note: I have created this writeup with only the symbols that would display on my browser. Every code above 255 in the html symbol reference displays as '?' regardless of whether or not I use the numerical of alpha representation. Therefore, please have mercy on my crappy looking symbols. Once I figure this out, I will do the writeup for topological transitivity.