Extracting Order from Chaos
Now, we will take a look at a cross section of the true phase space.
To do this we 'strobe' the system whenever the forcing term is at a chosen value. In this system the forcing term repeats itself with a period determined by the forcing frequency given to the system. Since the forcing term is dependent and periodic with the time term we can use the value of the system's time to get these points. For our example, the forcing frequency is 1.0, so the forcing repeats itself with period 2*pi. So, whenever the time variable is an integer multiple of 2*pi, we plot a point in phase space. This gives the cross section of the phase portrait at forcing term = amplitude*cos(n*2*p) = forcing amplitude. Sections corresponding to values of 2*pi plus a constant produce different plots, but the qualities of the system are much the same, even if the picture is very different, so we limit discussion to original plot of multiples of 2*pi.
This plot is called a 'Poincare section' of the phase portrait. Amazingly the system is bounded at any integer multiple of 2*pi, but it is bounded in a twisted mess of points. If we watch the section appear, the points hop erratically over the picture, but always stay bounded on the shape we see above. It is easy to see the lines in this system, but at finer detail, we see that these lines are made up of finer lines, which in turn are made up of more lines, and so on. The phase portrait of this system traces out what is called a 'strange attractor'. It can be shown that all trajectories stay on this attractor, even though they seem to hop erratically all over it. . This strange attractor seems to be embedded in the system, guiding the trajectories along it, and not letting them escape.
This plot was done with 1000 points , and because we use 100 iterations by drive cycle (one complete period for the forcing), it necessites 100 000