We found that it was not difficult to observe chaos in our pendulum system. Using a Pasco's Science Workshop program, we were able to read in data for the angular velocity and angular position of the pendulum. Unfortunately, the software available to us was not capable of taking data in sync with the driver, so we were unable to study Poincaré sections. This task is left for the mathematical model of our system and can be seen in the Mathmatical Modeling section. Phase portraits reveal much about the system.
A system may at first look chaotic but after some time oscillations will be periodic. Transients in the system must die out before we can determine whether or not a system is chaotic. We found that it often took over 10 minutes for transients to die out. Technically, we would have to let the experiment run for an infinite amount of time in order to check that the phase or other factors never repeated themselves to be absolutely, positively sure that what we were observing chaos. This would be impossible, obviously, so we must let the system run a while, studied the phase diagrams, and looked at the Fourier spectrum of the data. Below, a phase portrait (angular position vs. angular velocity) reveals an instance of such a transient "dying out". Notice the deceptively "chaotic" behavior it exhibits until suddenly its motion becomes periodic (the dark black ring in the middle of the graph). Most of our data was like this. We found chaos only in small intervals as shown later.
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Notice that the Fourier transform for non-chaotic behavior has a discrete number of peaks.
The disappearance of transients can also be seen, much more clearly in fact, in the following graph of theta vs. time (not from same data set as above). Erratic motion suddenly become periodic:
As discussed in the theory section, non-linear systems may sometimes oscillate at multiple amplitudes when the same parameters are imposed on the system. We understand this behavior to be an interplay between potential wells. For our system, the potential wells look like this:
Notice that there are essentially three separate wells in the above figure. Two small wells reside at the bottom and then there is the overall large one. Due to the nonlinearities of our system, we found it possible to have the system oscillate at two different amplitudes for a single frequency. Here, we have set the driving frequency to 3.61 rad/s which is 2/3 the resonant frequency for the system at the bottom of the small wells. For the driver arm length set at 4.15 cm, the pendulum oscillates in the right potential well in a non-chaotic manner.
Then, we gave the system a "kick" by simply bouncing the pendulum manually. None of the parameters were changed. When this happened, the pendulum began to oscillate through both of the potential wells with the amplitude of oscillation increasing by a factor of 10. The pendulum continued to oscillate in this manner and was unable to drop down into its original smaller oscillation unless we manually retarded its motion. Phase diagram for larger amplitude oscillation at driving frequency: 3.61 rad/s and driver arm length: 4.15 cm
We gradually increased the length of the arm (forcing coefficient) and found that the system was able to oscillate at these two different amplitudes until the arm length was 5.4 cm. At this point, the system was forced to oscillate through both wells as in the graph immediately above this paragraph. Again, take notice of the constant, periodic behavior with the phase history repeating.
Angular Frequency: 2.76 rad/s
We measured the resonant frequencies of the bottom potential wells to be 5.54 rad/s for left well and 5.625 rad/s for right. We expected that we would find chaotic motion if we drove the system at either of these frequencies. Here, we observed chaos when driving the pendulum at 5.51 rad/s. The data resulted in the following phase plot:
Angular driving frequency: 5.51 rad/s.
Notice the Fourier transform of the phase history has a smeared range of frequencies rather than discrete values. We find that oscillating the pendulum at this value, chaotic behavior occurs for driving arm-lengths greater than 5.32 cm. Below this value the transients eventually die out even though the driving frequency is very near resonant frequency This meant that the energy of the system was not great enough to escape the potential well.
After changing the parameters, we find chaotic motion occurs in pockets. We were unable to find pockets of chaotic behavior at driving frequencies greater than 5.51 rad/s.
Here we have what may look like chaotic behavior at 5.384 rad/s. From the Fourier transform, however, we see that the motion is composed primarily of one frequency. The phase history appears that it is close to repeating, so we conclude that this is NOT chaotic. Here, the driving frequency is closer to the resonant frequency of the wells, but we actually are not at chaos, as you might expect. This shows that there may be pockets of chaotic behavior in a range of initial parameters. There will not be one set interval where chaos will occur.
Lowering the frequency further, we find another pocket of chaos:
Angular driving frequency: 4.78 rad/s.
Again the erratic phase behavior and the smeared Fourier transform indicate that the motion is chaotic.
Here see chaotic motion at frequency 4.36 rad/s (within 20.5% of natural frequency). This is the lowest driving frequency at which we observed chaos:
Angular driving frequency: 4.36 rad/s.
We made the following observations about chaotic behavior: First, chaos occurred only where the total energy of the system was great enough to allow oscillation between two potential wells. We never saw chaotic motion when the system oscillated in just one well. Second, chaos was found most easily when the system was forced at or near the resonant frequencies for the bottom of the potential wells. Yet, this was not the only location where chaos was found. Chaos could be found in pockets, not in one range of initial parameters.