The Power Spectrum for this system allows for the determination of the various frequencies of the system.  It is created by taking the Fast Fourier Transform of the Time Series data and graphing as a function of Power.  The x axis is plotted by the number of bins, not the frequency, so the time step of the system must be taken into account. 

   

g   =         q   =        

Reinitialize

Observations:

(a)

(b)

(c)

Fig. 1:  Power Spectrum of the damped, driven pendulum when ωD = 2/3, g = 2, and Φ = 0.0.  (a) g = 0.9, the system is periodic; (b) g = 1.07, the system exhibits period doubling; (c) g = 1.15, the system exhibits chaotic behavior.

            For a periodic system, the resultant Power Spectrum should only contain one frequency which correlates to the drive frequency (ω).  Higher harmonics can also be visible for this system.  For a system exhibiting period doubling, the resultant Power Spectrum should display two frequencies for the system.  These frequencies should be found at ω and ω/2.  Period quadrupeling should exhibit three frequencies found at ω, ω/2, and ω/4.  This system is symmetrical, so there should always be an even amount of frequencies for the system.  A chaotic Power Spectrum should not demonstrate specific frequency peaks, but instead be found at a broad range of frequencies.