Math 145, Winter 1999: Chaos Theory

Lecture 1F, 08 Jan 1999
3. Trajectories of a vectorfield

Assume a one-dimensional state space, a straight line segment, and a vectorfield on it. We visualize the vectors lying right on top of the state space.

Each vector is now going to be interpreted as a velocity vector. Suppose that units have been agreed for measuring both space and time, say feet and seconds, and that speeds are given by the velocity vectors accordingly in feet per second. For example, let the vector at the point p indicate a forward velocity of 15 fps.

Orient the state space (blue) and the velocity vectors (green) vertically. Extend a new dimension (yellow) horizontally corresponding to time. Finally, we choose a time, t, for this construction.

Here is the space-time plane, showing the agreed units and the chosen point, (t, p).
Attach a small red rectangle centered at the point (t, p.) Choose its size so that: Draw a red diagonal line through the red box, from the lower left corner to the upper right corner. This diagonal line segment is the direction element determined by the green vector at p.

NOTE: if the vector at p had pointed down, we would have drawn this diagonal from the upper left corner to the lower right.

Here is the direction element at the chosen point.
This construction is repeated for all points (t, p) on the horizontal line through the fixed p as t varies. So all direction elements on this horizontal line are the same. Then this is repeated for a new point p, and so on.

To illustrate the process, we choose the open interval (0, 2) as S, and the logistic function as the vectorfield.

Here is a view of this vectorfield, shown in the tangent bundle representation.

And here is a plot of the direction field of this vectorfield, with trajectories superimposed.
Trajectories are curves, that is, graphs of functions in the space-time plane, which are tangent to the direction elements at all points.
Revised 09 Jan 1999 by Ralph