Chaotic structure in phase space

In the examples of the previous sections we investigated several cases of harmonic oscillators under different circumstances. In the exercises we have seen the behavior of the trajectories in phase space. Look at the plots you created, and you will identify the following characteristic structures:

• Limit cycles: ellipse-like figures with frequencies grater than $\omega _0$. There may be sporadic changes among the limiting cycles.

• Strange attractors: well-defined, yet complicated semi-periodic behaviors that appear to be uncorrelated to the motion at an earlier time. These are highly sensitive to the initial conditions. Even after millions od observations, the motion remains attracted to these paths.

• Predictable attractors: well-defined, yet fairly simple periodic behaviors that are not particularly sensitive to the initial conditions. These are orbits, such as fixed points and limit cycles, into which the system prefers to settle. If your location in phase space is close to a predictable attractor, ensuing times will bring you to it.

• Chaotic paths: regions of phase space that appear as filled-in bands rather than lines. The continuity between the bands implies some very complicated behaviors, yet the general motion still has some underlying structure.