Since the Time Series is unable to distinguish between small differences in the system (period-doubling), a more indepth approach is needed.  The Phase Diagram for this system is a mapping of the anglular velocity (ω) of the pendulum as a function of the angle (θ) at every point in time.

     

g   =         q   =        

Reinitialize

Observations:

          The following graphs are the Phase Diagrams that can be observed for the system for various values of the driving amplitude (g).  It can be seen in the above graphs, that for a periodic system, the resultant phase diagram for the system will contained on a single loop.  A system exhibiting period doubling will oscillate between two loops.  And a chaotic system will not be contained, but instead the phase diagram will encompass the  entire allowed phase space.

(a)

(b)

(c)

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

 

(a)

(b)

(c)

(d)

Fig. 2:  Additional Phase Diagrams for the damped, driven pendulum when ωD = 2/3 and q = 2.  (a) g = 1.35, the system exhibits periodic behavior; (b) g = 1.45, the system exhibits period doubling; (c) g = 1.47, the system exhibits period quadrupeling; (d) g = 1.50, the system is chaotic.