This image created with the
Poincaré Section algorithm
is based on the dynamical system characterized by the expression,
where x' is the first derivative of x with respect to time and x'' is the second derivative of x with respect to time.
The attractor here is chaotic. The domain of this image in the map of possible states for this system is, (1.3, 5.79)/(3.603, -4.85).
The parameter values are, a=7.5, ω=1, φ=0, b=1, μ=0.05.
This image is the cross section of a bundle of strands that make up the attractor for a double well potential system. We let the model run for about 115 days of model time, which is about 1.59 million cycles of the force applied to the system. At the end of each cycle we put a blue pixel on the screen, marking the system state at that time. In effect we slice the toroidal system attractor at time zero of each cycle. We colored all the points "outside" the attractor blue-green. The "inside points are the white ones.
Notice that the states fall in a distinctive fractal pattern. By comparing the three images in this row you can see that the pattern, this one, at the beginning or the attractor, 1/3 of the way around the attractor and 2/3 of the way around the attractor are not the same. The attractor gets flattened and folded at different parts of the cycle.