A Study in Chaos

Hamza Umar, Howard Community College
Natalie Paley, Howard Community College

Mentored by: Alex M. Barr, Ph.D.

Abstract

The onset of chaos is defined as the point where a system exhibits divergent behaviors when there are very small changes in initial conditions. For our experiment exploring the onset of chaos, we released a double pendulum from various initial angles. We used video recording equipment and software to accurately track the pendulum’s trajectory in each trial. Using these recordings, we measured the angular velocity of the bottom leg after five oscillations to determine the dependence of angular velocity on the initial release angle. A MATLAB simulation of our system was also used to investigate the same range of initial release angles. With the MATLAB data, we calculated the exponential increase in separation of trajectories with adjacent initial angles. In the experiment we determined that chaos first appears at initial angles above 50 degrees. The simulation showed the onset of chaos in the range of 48 to 50 degrees. The close agreement between the simulation and experimental results support a clear onset of chaos in this double pendulum system.

Introduction

     Science is built around understanding and predicting the behavior of various phenomena. Every system has initial conditions and natural laws which can be used to determine the behavior of that system. In non-chaotic systems, a small change in the initial conditions of a system leads to small changes in the behavior. In chaotic systems, however, a small change in the initial conditions leads to large changes in the behavior of the system. Measuring a system’s initial conditions exactly is impossible, and in chaotic systems a very small error in initial conditions leads to large errors when predicting the behavior of the system. For chaotic systems, the error grows exponentially over time, and thus, we lose predictability. This project demonstrates tools for pinpointing the onset of chaos in a system. Knowing where the limits of predictability lie can be useful in designing an engineered system in which the onset of chaos lies outside the operating range of the system.

     We chose the double pendulum for our project because it is relatively inexpensive to build in a machine shop or purchase commercially and it has initial conditions that can easily be controlled.  Previous studies on double pendula have experimentally confirmed chaos to be present for certain initial conditions. T. Shinbrot et al. showed that the double pendulum is chaotic at large release angles by comparing trials with similar release angles and showing exponential divergence between trajectories [1]. They used strobe photographs of the motion and measured the angles in each photograph by hand to track the trajectories. They did not however, investigate smaller angles to find the smallest release angle at which chaos first appears. More recently, T. R. Buresh et al. recorded the motion of a double pendulum in a video and tracked the motion in each frame using a computer program [2]. They investigated small release angles, below the onset of chaos, as they aimed only to understand the non-chaotic motion.

     Our project builds most directly on the work of R. Ø. Nielsen et al. They recorded videos of a double pendulum for many initial release angles and identified the minimum release angle at which chaos occurs [3]. In our project, we also identify the minimum release angle for chaos, however for a different class of initial conditions than those examined by Nielsen. We use a method similar to Nielsen to identify the onset of chaos, however, we modified their method to not require precise knowledge of the moment the pendulum was released.

Materials and Methods

     The double pendulum for these experiments was commercially supplied, and made of electroplated aluminum [4]. The upper and lower legs of the pendula are of equal width (3.7 cm), equal thickness (approximately 6.5 mm), and the upper legs are 21.4 cm long while the lower leg is 17.9 cm. With the volume and density values now determined, we were able to calculate the mass of each leg. The mass of the upper portion was calculated at 266 g and the lower leg at 116 g. These calculations exclude the connecting bolts and spacers. The angular position of the upper leg is represented by θ₁ and the lower leg by θ₂.  The first derivative of theta is the angular velocity, similarly represented by ω₁ for the upper leg and ω₂ for the lower.

Figure 1: On the left, the pendulum viewed head on, showing the two bars of the upper leg and single bar of the lower. On the right, a diagram showing the two θ values and the bullseye used for tracking the lower leg. Each leg has its own reference axis and angular position (which varies as the pendulum swings).

     Experimental trials were recorded by cell phone at a definition of 720p and frequency of 240 frames per second. Experimental data were analyzed using the free program, Tracker [5]. The bolts at the pivot points of the pendulum were used for measuring the angles of each leg and tracking the trajectory of the bottom leg. Each video was run through Tracker three times. The first time, the coordinate axis for the upper leg was mapped and the initial release angle (θ₁) recorded. Then using Tracker’s autotracker function, the coordinate axis for the angle of the second leg was mapped to the connecting bolt between the upper legs and lower leg. Finally, the autotracker was used to measure the angle of the lower leg (θ₂) to the moving coordinate axis through the course of the pendulum’s trajectory (see Figure 2). An orange and white bullseye was taped to the end of the lower leg to provide a distinguishing feature for the autotracker to map in each frame (see Figure 1). The angular velocity was calculated in Excel using two successive angular positions (Equation 1). To estimate the limitations of autotracker, an uncertainty test was conducted by analyzing a single trial multiple times with small variations in the axis position. The results of this test showed a difference of +/-1 degree in the angular position values over time.

Figure 2: The double pendulum with coordinate axis for the bottom leg and three seconds of motion marked by Tracker. Tracker traces the motion of the bullseye marked in orange and white near the bottom of the lower leg.

Figure 3: Two nearly identical release angles with increasing separation between trajectories. Initial θ 1(upper leg angles) are 70.2 degrees (solid curve) and 70.3 degrees (dashed curve). Initial θ 2 ,   ω 1 ,   a n d   ω 2 were kept constant at 0 degrees (hanging straight down) and 0 deg/sec for both releases. The bottom leg makes several full rotations, which is why θ 2 values go above 360 degrees.

     The onset of chaos is when the pendulum becomes highly sensitive to small differences in the initial release angle as illustrated in Figure 3. Small differences grow exponentially for chaotic trajectories. The angular velocity of the bottom leg at the fifth θ₂ = 0 crossing was used to determine sensitivity to initial conditions. Chaos can be found in the trajectories of a double pendulum by comparing similar initial conditions and observing the difference between the angular velocity values at the fifth θ₂ = 0 crossing. If the angular velocity values vary smoothly with varying initial conditions, the region of initial conditions is non-chaotic. If the behavior of the angular velocities changes erratically for small changes in initial conditions, chaotic behavior has been observed. We chose θ₂ = 0 as our crossing point because it will always be crossed by a pendulum with any amount of energy.  We chose the fifth crossing because it allows enough time for the separations between the trajectories to become measurable, while also being early enough to prevent damping from becoming an influencing factor.

    Initially, angular position was measured at t = 2.5 seconds after the pendulum was released. This approach of measuring position at a specific moment in time was similar to the Nielsen experiment [3]. We released the pendulum by hand and the start of motion was determined visually in Tracker. However, this methodology introduced error: even tiny differences in determining t = 0 (the moment when the pendulum was released) will cause time-based comparisons to be comparing different points in each trajectory. Our revised approach of focusing on the fifth θ₂ =0 instead of a specific moment in time ensures that we compare equivalent points in each trajectory.

     The experimental measurements were complemented by a simulation of the double pendulum in MATLAB. A second order Runge-Kutta algorithm was used to numerically integrate the equations of motion for the double pendulum and investigate the divergence of trajectories with neighboring initial conditions [6]. The separation between two trajectories (“a” and “b”) is calculated using both of their angles and velocities in the standard distance formula

For non-chaotic trajectories, the separation between the trajectories remains approximately constant as the trajectories evolve. For chaotic trajectories, the separation grows exponentially as the trajectories evolve. This behavior is illustrated in Figure 4.

Figure 4: Separation (Equation 2) vs. time for two trajectories with initial angles θ 1= 20 degrees and 20.1 degrees (solid blue curve) and with initial angles θ 1= 70 degrees and 70.1 degrees (dashed red curve). The exponential growth seen in the dashed curve indicates chaos. Trajectories differed only in the initial θ 1value. Initial values for θ 2 ,   ω 1 ,   a n d   ω 2were all zero.

     The Lyapunov exponent (λ) describes the exponential divergence of two nearby trajectories (see Equation 3). The larger the exponent, the faster the separation between trajectories grows.

     In MATLAB, Lyapunov exponent values were determined by numerically integrating the equations of motion for two neighboring initial conditions and graphing 1 t l n ( d / d 0 ) as the trajectories evolve. The value that 1 t l n ( d / d 0 ) converges to is defined as the Lyapunov exponent value. Chaotic trajectories are defined as trajectories for which 0″ src=”https://howardcc.instructure.com/equation_images/%255Clambda%253E0″ alt=”LaTeX: \lambda>0″ data-equation-content=”\lambda>0″ data-mathml=”“>λ > 0  while non-chaotic trajectories (with minimal damping) have λ ≈ 0 [7].

     For our experiment, we varied the release angle for the upper leg from 14.9 degrees to 93.9 degrees; the lower leg was always released aligned with its vertical axis at 0 degrees. For both the experiment and the MATLAB simulation, the release angle was changed slightly between each trial (half a degree for the simulation, and as close to the same degree as possible in the physical experiment). Non-chaotic initial angles do not exhibit large separations with time. Release angles with great separations between adjacent trajectories are, by definition, chaotic.

Results

     Below θ₁ = 50 degrees, releasing the double pendulum from similar initial angles results in very similar trajectories, giving similar angular velocities after the fifth θ₂ = 0 crossing. For release angles greater than 50 degrees, small changes in initial conditions result in significantly different angular velocities after the fifth θ₂ = 0 crossing. In Figure 5, the angular velocities are well grouped prior to 50 degrees, increasing in nearly a straight line as more energy is added to the system in the form of potential energy as the upper leg is released from larger heights with increasing initial angle. At angles greater than 50 degrees, the angular velocities show no correlation with the initial angle. The double pendulum becomes highly sensitive to the initial angle above 50 degrees. This can be seen in the large spread between angular velocities in nearby initial angle trials. The trajectories with release angles above θ₁ = 50 degrees exhibit angular velocity values that are scattered and have no apparent pattern, which suggests that the trajectories are chaotic.

     To test our interpretation of the initial angles greater than 50 degrees as chaotic, we modeled the double pendulum in MATLAB to calculate the divergence between nearby trajectories and determine the Lyapunov exponent. A positive value for the Lyapunov exponent indicates the presence of chaos, while an exponent ≈ 0 indicates non-chaotic motion. The first positive exponent value appears at approximately 48 degrees with a significant increase in exponent value occurring near 50 degrees (see Figure 6). Above 50 degrees, the system becomes increasingly chaotic except for brief dips in the Lyapunov exponent value around 53-56 degrees, 65 degrees and 79 degrees.

Figure 5: The angular velocity of the lower leg at the fifth θ 2 = 0 crossing versus the initial release angle of the upper leg. Angular velocity values for initial θ 1values below 50 degrees show a traceable growth. Above 50 degrees we see no apparent pattern in the angular velocity values.

Figure 6: The Lyapunov exponent versus the initial release angle of the upper leg. The simulation shows that the Lyapunov exponent is positive for the first time around θ 1 = 50 degrees, which represents exponential growth in separation and thus, chaotic behavior. This is also the case for most initial angles above 50 degrees.

Discussion and Conclusions

     We investigated the onset of chaos by calculating the angular velocity of the lower leg of the double pendulum at the fifth crossing through θ₂ = 0 for a range of θ₁ release angles. This was a modification on Nielsen’s approach, which examined the θ₂ angle 10 seconds after the pendulum was released. The modification removed the source of error associated with identifying the exact moment the pendulum was released. The plot of ω₂ vs θ₁ for the fifth crossing showed a sudden extreme sensitivity to initial conditions around 50 degrees. We identify this as the onset of chaos. This result was supported by the simulation results, in which the Lyapunov exponent becomes positive around 48 to 50 degrees as well. This means that beyond a release angle of 50 degrees, angular position and velocity can no longer be accurately predicted for this double pendulum system.

     A direction of further inquiry would be to perform more experiments at and around the angles in the simulation where the Lyapunov exponent value suddenly decreases (around 53, 65 and 79 degrees). These regions may provide clues as to how to control or alter the degrees at which chaos occurs.

     To expand upon this experiment, we could also modify the moment of inertia of the lower leg to see how this affects the onset of chaos. The moment of inertia is dependent on the length and mass distribution of the legs of the pendulum and can be modified by changing the lengths or adding various masses to the leg. The efficiency of energy transfer between the upper and the lower legs is dependent upon their natural frequencies which depend on the moment of inertia values. If we vary those frequencies, we can cause resonance which will amplify the rate of energy transfer and the onset of chaos might appear at smaller release angles.

Acknowledgements

     We thank Mikiyas Bokan for his contributions to this research. His creation of the MATLAB simulation and Lyapunov exponent graph were invaluable. This material is based upon work supported by the National Science Foundation under Grant No. 1458149. Any opinions, findings, and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Contact: natalie.paley@howardcc.edu, hamza.umar@howardcc.edu

References

[1] T. Shinbrot, C. Grebogi, J. Wisdom, J. A. Yorke, Chaos in a Double Pendulum, Am. J. Phys., 60(6), pp. 491, 1992.

[2] T. R. Buresh, D. Aliaga, and M. J. Madsen, Chaos of the Double Pendulum, Wabash J. Phys., 2011.

[3] R. Ø. Nielsen, E. Have, B. T. Nielsen, The Double Pendulum, Niels Bohr Institute, 2013.

[4] We use a double pendulum purchased from 3B Scientific, https://www.a3bs.com/chaotic-pendulum-e-1017531-u8557340-3b-scientific,p_576_25075.html.

[5] Tracker Video Analysis, https://physlets.org/tracker/. Version 5.0.6.

[6] A derivation of the equations of motion for the double pendulum is available at https://tinyurl.com/DblePendEOM

[7] L. Reichl, The Transition to Chaos 2nd Ed. (Springer-Verlag, New York, 2004).