Lorenz Attractors and Trajectories

[EbolaPox]

Homework Statement

I'm using the textbook "Nonlinear Dynamics and Chaos" by Strogratz. So far, it's been terrific. However, I got to the section on the Lorenz attractor and got stuck. The author has a picture and an explanation that eludes me. The picture in question can be seen on this google books page (page 320 , use the search bar and type Strange Attractor). The paragraph directly below figure 9.3.3 in which he states "the pair of surfaces merge into one in the lower portion of Figure 9.3.3". What pair of surfaces is he referring to in that picture? It just looks like part of the trajectory. What do the dots represent in that picture?

http://books.google.com/books?id=FIYHiBLWCJMC&printsec=frontcover&dq=Strogatz+Chaos#PPA320,M1

To make this make a bit more sense, on the page before, he introduced a picture of a trajectory that Lorenz originally discovered. That picture is the same as the top-right most picture on this wikipedia page:
http://en.wikipedia.org/wiki/Lorenz_Attractor

The first picture there on this wikipedia link is a trajectory. The author then starts talking about lorenz attractors. I'm not too sure I understand the connection. I thought attractors were points that trajectories stayed near, and now it seems like he is calling the trajectory an attractor. Is he implying that this weird trajectory Lorenz discovered goes to an attractor or to a set of attracting points?


What is the difference between a lorenz attractor and the trajectory from solving the Lorenz equations? Is it the set of points that the trajectory tends towards as t-> infinity, or is it the trajectory itself? Is there just one Lorenz attractor for certain parameter values of the Lorenz equations and different initial conditions lead give us trajectories that tend to a Lorenz attractor?

Sorry for all the questions, and thank you very much for any help.

 

[danb]

"the pair of surfaces merge into one in the lower portion of Figure 9.3.3". What pair of surfaces is he referring to in that picture?

The "ears" or upper lobes of the picture. Paths in those areas merge with each other in the lower region.

What do the dots represent in that picture?

Nothing in particular. They're just areas that contain some significant density of paths.

I thought attractors were points that trajectories stayed near

An attractor doesn't have to be a point (0D). For example, a limit cycle is a loop-shaped attractor (1D). That entire picture is the attractor for the Lorentz oscillator. It's a bounded, irregular orbit with a noninteger (fractal) dimensionality (~2.05D).