RE: What Is Chaos Theory?

Attractors and detractors in chaos theory are intimately related to fixed points. In fact, fixed points and detractors / attractors are the same.

What changes is the fundamental behavior that occurs around each fixed point.

In 2-space, this can be catalogued with the derivative of the iterative function F at the point x. The slope of the tangent line and its magnitude determine two properties : attraction and spirality. This can also be characterized by eigenvectors and eigenvalues of a dynamical system, if you consider a dynamical system in discrete updates as being the results of the iterative function, and the iterative function being an update function, like Euler's method, or any other such choice.

In 3 (and higher) spaces, the type of fixed point, be it an attractor or detractor, depends on the direction at which the fixed point is approached in space.

A fixed point may be both repelling and attracting, but how you approach, i.e. the direction that is chosen, the fixed point determines which of these it is.

Attraction is related to a fixed point's eigenvalues in that if all the real parts of the eigenvalues are negative, then the fixed point is an attractor.

But if some eigenvalues are negative and some are positive, one needs to consider the path that approaches the fixed point.

One does this by considering the so-called eigenbasis, i.e. a basis of the space that is given by the (generalized) eigenvectors.

Now, locally, you can re-orient the frame about the fixed point in terms of a different basis. Just take your right-hand and do the right-hand rule with it. Now, the thumb and the index and middle fingers form the 3 different vectors in 3-space. If you slightly deform them, you can make the vectors not orthogonal to one another. And then you can also rotate your hand in any way. The vectors that your 3 fingers are pointing in are still a basis of 3-space, and if you think of the centroid of those three fingers as the fixed point, you are then considering a basis about that point.

Now, consider, for simplicity, any smooth path towards a fixed point. If we look at the direction of the tangent line at a particular moment in time on our path, we can then rewrite this vector in terms of our eigen basis. The weight of each eigenvector, i.e. the corresponding coefficient in the linear combination, tells us to what extent that eigen vector plays a role in the current direction that is used for the approach to the fixed point in question, which in turn then tells us how much attraction / detraction that is happening for that path. This is done by looking at the eigenvalues of which eigenvectors have the most 'weight' for a particular apprach.

In other news, if one of the eigenvalues is complex, one sees spirality in the direction of the eigenvector that corresponds to that complex eigenvalue.