The Time Series allows the user to observe the angular velocity (ω) of the system as a function of time. 


g   =         q   =        



If the system is driven at a small driving amplitude (g = 0.9), the system will be periodic.  After a short period of time, the transient effects of the system will die out and the resultant steady-state will be periodic (1a).  At this point, the frequency of the pendulum will be equal to the frequency of the driving force.  For every complete swing of the pendulum, the driving force completes one cycle.

            If the system is driven at a slightly greater driving amplitude (g = 1.07), the system will exhibit period doubling.  After the transient effects have died out, the steady-state of the system will appear to be periodic but with closer examination can be seen to have two different frequencies (1b).  Fig. 2 is a magnified section of Fig. 1b and enables us to see the difference in the two peaks.  The motion of the pendulum is also different in this system because for every complete swing of the pendulum, the driving force goes through two cycles instead of one cycle as in the periodic state.

            If the amplitude of the driver is increased further (g = 1.15), the system will become unstable and chaotic.  Even after long periods of time, the system will never reach a periodic state.




Fig. 1:  Time Series for the damped, driven pendulum when ωD = 2/3, and q = 2.  (a) g = 0.9, the system is periodic; (b) g = 1.07, the system exhibits period doubling; (c) g = 1.15, the system is chaotic.


Fig. 2:  Examination of the differences in heights of peaks for Time Series when ωD = 2/3, q = 2, and g = 1.07.