Chaotic Spring Pendulum System

Joe Marino and Chris Nolting

University of Minnesota

Methods of Experimental Physics Spring 2013

Our project was based off of a paper by Robert DeSerio. All references used are listed at the bottom of this page. Experimental methods and techniques are described below. Also included are ideas for future MXP projects that would be able to continue on and go further.

Data Collection & Analysis Techniques

The initial stage of data collection consisted of performing a series of trials while varying the amplitude and frequency of the stepper motor. We mapped out the chaotic and periodic regions of the system by taking one-minute runs at five driving arm amplitudes in the range of 1.54 cm to 5.56 cm. At each amplitude, we collected data over the driving frequency range of about 4 rad/sec to 8 rad/sec in about 0.1rad/sec intervals. Our LabVIEW interface collected data from the rotary encoder at a rate of 500Hz, giving us a total of 30,000 data points for each one-minute trial. The stepper motor’s resolution of 0.9° indicates that there are 400 discrete stepper motor angles. These 400 angles over our 30,000 data points means that each Poincare section had around 75 points, which was a sufficient amount to determine whether the behavior was periodic or chaotic. Assembling the behavioral information from each Poincare section provided us with a bifurcation diagram of the system. With our qualitative mapping complete, we could then go on to explore chaotic and periodic regions of interest by finding their Lyapunov exponents.

The Lyapunov exponents are a measure of the system’s divergence over time. Based on previous experiment (7), we initially thought we would need to do overnight data runs to get sufficient data to fit, however it quickly became obvious that after even a few short seconds that the two similar trajectories were no longer coupled. Therefore, we decided to take shorter data runs (30 seconds) with a higher sampling frequency (10kHz). Measurements consisted of setting up the system at specific pendulum and driving motor angles and letting it run for 30 seconds. Next, we ran the system again with the same driving amplitude and frequency and initial driving motor angle, but with an initial pendulum angle that differed by 2° from the first run. This small difference in angle was measured by placing two dots behind the metal plate of the pendulum then using a small hole in the plate to align the pendulum with the appropriate dot. With this method, we were able to align the pendulum to an accuracy of about 0.5°. We started the driving motor at an angle of 0° for each run to assure that the only difference in phase space was in the angular position coordinate. We chose to only investigate the Lyapunov exponents of the angular position based on the fact that it is difficult to accurately start the pendulum at a desired initial angular velocity. To find the Lyapunov exponent, we compared the data from our two similarly initial conditioned runs by finding the difference in angle over time. This difference, D=θ2-θ1, oscillates with an amplitude growing exponentially with time, as shown in figure 8 below.

Figure 7: Plots of two trajectories separated initially by 2°. By 7 seconds they are no longer ‘coupled,’ meaning they do no longer follow similar trajectories and are too far displaced from each other in phase space for our linear approximation to hold. Driving amplitude: 4.63 cm Driving frequency: 5.18 rad/sec

Figure 8: The difference between the two trajectories from figure 7 vs time. Starting from 2 degrees difference, the difference between them oscillates and the amplitude of this oscillation grows exponentially with time due to parametric resonance.

To find the Lyapunov exponent, we fit the peaks of this oscillation to an exponential. We found the upper peaks of the oscillations for times at which the two trajectories were still coupled and kept only those points. From the plot in figure 8, we estimated the uncertainty in time as being ±0.02 sec and the uncertainty in angular difference as being ±0.25°, the resolution of our rotary encoder. Knowing our initial angular difference, D₀, we plotted vs. time, which gave us the Lyapunov exponent as the slope of the resultant linear fit. We did this for multiple points on the boundary between the periodic and chaotic regimes for our system. In this paper, we have reported the results for a driving amplitude of 4.63cm and a driving frequency of 5.18rad/sec as an example case.

Numerical Simulation

Using Matlab, we generated a numerical simulation based on equations 2 and 3. We entered our measured physical parameters from table 1, but we were not able to directly measure the quantities b or b’. Initially, we used some “best guess” values for these parameters, based on the values used by DeSerio (4). We then were able to run simulations and generate similar data sets to those we had previously acquired as described in the Data Collection section. Running the same types on analysis on these simulated data sets, we were able to make comparisons between the phase space diagrams and the calculated Lyapunov values to see how well our simulation matched our observations of our system. The parameters b and b’ were then adjusted, along with a few of the more uncertain parameters, such as the spring constants and the moment of inertia of the system. Finding the set of parameters that made our simulation best fit the observed data gave us better values for b, b’, and our spring constants and the moment of inertia. We were not able to perfectly match the observed data to our simulation by adjusting the input parameter values, however.

Figure 9: Plot of data and simulation for a trial at driving amplitude of 5.56cm and driving frequency of 5.23rad/s. The simulation follows the data up until around 8 seconds. A better set of system parameters or perhaps a different model is required to better predict observations.

As can be seen from figure 9, our simulation was able to qualitatively match our observations, however, it was unable to perfectly follow observation. This is due to the nature of chaotic systems, and how our uncertainty in our system parameters compounds over time and eventually our simulation diverges from the true trajectory. Using our simulation, we were able to locate the boundary between the periodic regime and chaotic regime for our simulation and plot that onto our bifurcation diagram as a comparison.

Results and Conclusions

As mentioned in the previous section, the behavior of each data run was determined by plotting a Poincare section for an arbitrary fixed motor angle in phase space. Periodic Poincare sections were characterized by two clusters, each spread only within the angular position and angular velocity uncertainties. The chaotic Poincare sections were characterized by a large spread of points without well-defined clustering. Sample Poincare sections are shown in figure 9 below.

Figure 9: Samples Poincare sections indicating periodic (left) and chaotic (right) motions. The left is at motor amplitude 2cm and driving frequency of 560 Hz. The right is at motor amplitude 5cm and driving frequency 700 Hz.

It should be noted that while for most Poincare sections, a clear distinction could be made, not all resembled those in figure 9. For some sections near the boundary between regimes, there was a mixture of characteristics, with some points in a dense locus and other points spread out. This required us to impose a limit: if there was more than 20° spread in angle or 50°/sec spread in angular velocity, the motion was classified as chaotic.

The bifurcation diagram we generated by looking at the nature of the Poincare section is a topographical mapping of the periodic and chaotic regions of our system. The bifurcation diagram can be seen in figure 10 below, with both observed natures and the boundary between regimes plotted from simulation data.

Figure 10: The bifurcation diagram of our system. In blue are periodic regions and in red are chaotic regions. This determination was made from the Poincare sections of 1 minute runs. The boundary between regimes from simulation is also plotted.

From this diagram, one can see that for certain small driving amplitudes, chaotic behavior is not observed. Also, once the driving force is large enough to create chaotic behaviors, there is an upper limit for the driving frequency in the chaotic regime, above which the motion becomes periodic again. Also note that for larger driving amplitudes, the chaotic region has a larger extent in driving frequency. A very interesting characteristic was the presence of “islands” of periodic behavior embedded within the chaotic “sea.” There was not enough time to closely examine these areas to better understand why this might have occurred.

By plotting ln(D/D_0) vs. time, we were able to generate a linear fit of the peaks of the oscillating difference between two similarly initial conditioned trajectories in order to find a value for the Lyapunov exponent of the system. While we did this for areas near the boundary between periodic and chaotic regimes, in this paper we only quote the results of one case (Driving Amplitude = 4.63cm and Driving Frequency = 5.18rad/sec). If we find our linear fit to be ln(D/D_0)= a+b*t , then our exponential fit is given by D=D_0*e^a*e^(bt). In figures 11-13 below, the linear fit and resulting exponential are shown in detail.

Figure 11: A linear fit of vs t. The fit equation is ln(D/D_0 =(0.2261±0.0081)*t+(0.416±0.043)). This gives a Lyapunov exponent of 0.2261±0.0081 s^-1 and an initial angle difference of 3.03±0.13°. The reduced value for the fit was 2.04.

Figure 12: A plot of the difference D vs time overlaid with the exponential calculated from our fit.

Figure 13: A plot of the χs for the fit.

The fit equation generated was ln(D/D_0 =(0.2261±0.0081)*t+(0.416±0.043)). From this, the Lyapunov exponent was λ=0.2261±0.0081 s^(-1). As can be seen from the χ plot in figure 13, the residuals appear randomly distributed and none are much larger than 2. The reduced χ² for the fit was 2.04, implying the linear fit is pretty good. We performed this same fitting procedure on simulated data taken at the same driving frequency and amplitude. The simulated Lyapunov exponent was found to be 0.184±0.009 s^-1, which at least matches the order or magnitude of our measured Lyapunov exponent of 0.226±0.008 s^-1. These values also qualitatively resemble those of DeSerio (4) who got Lyapunov exponents between 0.42 s^-1 and 0.62 s^-1. Our value is different is due to differences in driving frequencies and amplitudes and differences in the physical parameters of the system.

Our observation that chaotic trajectories diverge very rapidly (over a span of a few seconds) and our calculated Lyapunov values call into question the work of a previous MXP experiment (7) which took very long overnight data runs and found a Lyapunov exponent value of λ = 0.297±0.044 hr^-1. While the group did study a lower range of driving frequencies, based on our experiment, it seems unlikely that any initial information would be retained after hours of data collection.

Future Directions

Due to time constraints and spending too much time finding solutions to data collection issues that arose and unexpected complications with the Lyapunov calculations, we did not have time to get to some interesting things that could be done with this project. Future groups could go further and explore the things that we did not get a chance to. With more time, we would have looked into the bifurcation space more closely. It would be very interesting to be able to more exactly determine the boundary between periodic and chaotic regions. This would also include examining the “islands” which had periodic Poincare sections in an area surrounded by chaotic points. It would be good to find these areas and their boundaries and look for a theoretical reason for why they occurred.

Another aspect of our project that we were initially interested in pursuing was the use of variable driving motor frequencies. We wanted to try variable frequencies of the form Ω(t)=a sin (bt)+Ω_0 with as well as small amplitude Gaussian noise around a constant frequency value. For the variable frequencies, the value of b would be kept at 1Hz so that the effect of the modulation would be slow enough to be observed. The reason we thought this would be interesting would be that since the system’s motion was so sensitive to the exact conditions and parameters, effects due to the noise intrinsic in the system could be studied. This intrinsic noise is inescapable in any apparatus and accounts for the random fluctuations in the experimental environment that occur throughout the measurement period. We would find how the variable frequencies and noise affected the boundaries of the bifurcation diagram and how they affect the Lyapunov exponents of the system.

Finally, it would be very interesting and helpful to find a way to better match the simulation and the observed data. It could be that we just were not able to find the correct set of parameters to represent our system, however after painstakingly testing and changing parameters, we began to think that perhaps our model in equation 2 is not perfect at modeling all of the effects on our system. There could be other terms that are important that we are not accounting for, or some slight flaw in the way we’re modeling one of the effects already present. Immediately, we can think of two possible flaws. First, we doubt the axel damping term which depends only on the direction of the motion. This is quite possibly an over-simplification that is introducing and error that causes us to not be able to match the observed data. Another simplification that was used in the modeling was assuming the stepper motor was a continuous driving force, when it really moved in discreet intervals. While these intervals were very small so the motion of the motor appeared continuous, there could be effects due to its discreet motion.

General Advice for Future Projects

We spent too much time programming LabVIEW to control our stepper motor. Using the function generator to create the signal works just as well if you are not trying to use variable frequency signals. Also, two function generators can be used in combination to create a modulated frequency signal like the one described above. Simulations are valuable, and become comfortable with using Matlab to generate them, create plots, and manage large data sets. Excel cannot handle data sets with more than about a million entries and if you are sampling fast over a long period of time, this can be a problem. Finding a better way to accurately measure parameters and fit the simulation to the observation would be very valuable and greatly improve upon our work.

References

(1) Strogatz, S.H. Nonlinear Dynamics & Chaos. Reading, Massachusettes. Perseus Books. 1994

(2) Motter, Adilson, and David Campbell. "Chaos at Fifty." Physics Today. 66.5 (2013): 27. Web. 11 May. 2013. http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1.

(3) Gleick, James. Chaos, Making A New Science. Vintage. 1987

(4) DeSerio – Am. J. Phys. Vol. 71 No. 3 March 2003 pg. 250-257

(5) Illinois State University – Department of Physics. Physics 320 website. http://www.phy.ilstu.edu/~rfm/320S09/GN2_PendPhsPort.html

(6) Gitterman, Moshe. The Chaotic Pendulum. New Jersey: World Scientific, 2010.

(7) Clasen & Glaser. “Chaotic Pendulum.” Methods of Experimental Physics (2011): n.pag. Web 2/20/13 http://mxp.physics.umn.edu/s11/Projects/S11_Chaos/.