Chaos and Back again
Route to Chaos
Depending on parameters like frequency and amplitude of the driving sled, the pendulum can show up different behaviour. Typical "pre-stages" of chaos are period-doubling, period tripling etc. orbits, which are shown below. The sound you can hear and which is displayed in the spectra below corresponds to the angular speed signal of the pendulum, with time scaled by a factor 1000.
Sine of driving position, corresponding to blue spectral lines below
Speed signal in period doubling case. An octave BELOW the driving signal above!
Period Doubling Spectrum
Speed signal in period tripling case. A duodezim BELOW the driving signal above!
Period Tripling Spectrum
Chaotic Pendulum. Speed Signal shows a 1/f- Noise behaviour
Spectrum of speed signal in chaotic case (driving frequency (blue) higher than in examples above)
Transient Chaos
For some parameter sets the strange "attractor" is no attractor anymore (yet an invariant set of the flow). After some (unpredictable) time the system switches back to a stable periodic orbit.
Transient (i.e. instable) Chaos
Spectrogram for transient chaos: System switches to stable periodic orbit
