The connection with the driven pendulum is illustrated by the association of the onset of chaos in this system with the onset of nonmonotonic behavior in the one dimensional return map. The pendulum equation is:
For a=1, b=1.576, c=1.76, d=1.4 the motion is quasiperiodic and the return map is smooth and monotonic. Changing the damping to b=1.081 and the running force to d=1.094 (to keep the ratio of the two frequencies roughly constant at 0.38) gives chaotic motion, and the return map develops a minimum (around x_{n}=0.75 for the Poincare section used here) and "kinks" in the curve (e.g. expand the region around x_{n}=2.2, x_{n+1}=3.6).
a | b | c | d | Dynamics | Power Spectrum | Return Map |
1 | 1.576 | 1.76 | 1.4 | |||
1 | 1.253 | 1.76 | 1.2 | |||
1 | 1.081 | 1.76 | 1.094 |
You can also look at the Poincare section to compare with the 2D Circle map and follow the "crinkling" of the invariant circle. Note that the dissipation rate is larger than those used in chapter 2 so the fractal nature of the section is less evident.