The Bifurcation Diagrams allows a comparison between the periodic and chaotic behaviors of the system.  For this graph, the angular velocity (ω) is plotted as a function of the driving amplitude (g) for a given angle of the drive cycle.

q   =

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Observations:

Below are some bifurcation diagrams that can be obtained for various values of the dampening factor (q).

 (a) (b) (c)

Fig. 1:  Bifurcation Diagram of the damped, driven pendulum when ωD = 2/3 and Φ = 0.0.  g ranges from 0.9 to 1.50  (a)  q = 2; (b) an expansion of (a); (c) q = 4.

Depending on the initial conditions of the system, the behavior can follow different branches to chaos.  Since the Bifurcation Diagram is viewed stroboscopically, a periodic system will have one point.  A system exhibiting period doubling will have two points and a chaotic system will have multiple points.  Figure (b) is an expansion of the bifurcation diagram for the system when experiencing the first bifurcation into period doubling and then chaos.

For the figures seen in the above figures, the angular velocity is taken when φ = 0.  The transient effects are accounted for by omitting the first 15 drive cycles.  For a system with a dampening parameter of 2, when driven at a small driving amplitude (g ~ 1), the motion is periodic and the pendulum oscillates at the same frequency as the driver.  The resultant Poincaré Section of this system is a single point.

If the driving amplitude is increased further, the oscillations split and have frequency components of ωD and ωD/2.  This is apparent in figure (b).  This figure also allows us to see that before the system reaches chaos, the frequencies will bifurcate once more to exhibit period quadrupeling.  The resultant Poincaré Section of this sytem would have four points.

Areas in the bifurcation diagram that contain an infinite number of points for a single value of the driving amplitude, are areas of chaos.  Within the large areas of chaos there can be seen small instantaneous periods of periodicity.  (g ~ 1.12).  For this reason, the chaotic areas of the system are very complex.

Past the chaotic region of figure (a), it can be seen that the system once again reaches a periodic state.  Although the system appears to be experiencing period doubling, but at this point, the system looses its symmetry about the origin and can have two different shapes depending on the initial conditions.  This region is initially periodic and can be seen to enter a new cascade into chaos.

If the dampening parameter of the system is reduced to q = 4, then the resultant effect is that the regions of chaotic behavior are much broader.  Figure (c) is the bifurcation diagram for this case.  It can be seen that the regions of chaos are indeed much broader as well as the periodic behavior found at g = 1.25 is much more prominent.