Chaos of the Double Pendulum - Wabash College: Academics Physics … · · 2011-12-16WJP, PHY381...

WJP, PHY381 (2011) Wabash Journal of Physics v2.3, p.1

Chaos of the Double Pendulum

Y. Tang, E. Groninger, B. Foster, D. Aliaga, T.R. Buresh, and M.J. Madsen

Department of Physics, Wabash College, Crawfordsville, IN 47933

(Dated: December 16, 2011)

We characterized the chaotic motion of a damped physical double pendulum which

has not been achieved previously. The results consist of scattered phase plots, fractal

Poincare sections, and positive Lyapunov exponents.


The model double pendulum system of point masses is known to be one of the simplest

systems to describe chaotic motion [5]. The signature of chaos is exponential sensitivity

to initial conditions. [2, 5, 6]. Previous work on measuring the chaotic properties of a

physical double-pendulum includes stroboscopic measurements of the motion over short

time scales, an offset distributed mass system, and a driven double pendulum apparatus

[1, 4–6]. However, only the driven oscillator system has been experimentally characterized

using the canonical measures for a chaotic system: a positive Lyapunov exponent, scattered

phase-space plots, and fractal Poincare Sections [1, 5, 6]. Prior experiments in driven double-

pendulum characterization have been performed with relative ease in measurement because

of the consistent data run ability. The natural physical double-pendulum is a non-driven,

damped system and thus requires many trial runs to achieve the amount of data needed

for the Poincare Sections [1]. Also, the rapid motion of each pendula made tracking the

angular position of each bar difficult. Stroboscopic measurements have been employed but

only allow for roughly ten points per data run, which resulted in Lyapunov exponents with

high uncertainties[6].

In this letter, we report the advancements made to previous experiments in the chaotic

double-pendulum system. Many of the past experiments’ focus was to characterize individual

signatures of chaos. We, however, attempted to preform a more rigorous analysis of the

double-pendulum’s chaotic motion. We employed a compound (distributed-mass) double-

bar pendulum for experimentation and developed an robust algorithm to handle the heavy

computations in the analysis. A high speed camera supplemented with computer software

solved the problem of large angular position tracking. We not only present the data and

analysis that fully characterizes the system’s chaos, but also propose a technique that is

generalizable for a larger class of previously unexplained chaotic systems.

Our pendulum consists of two 1/2 inch thick and 3/4 inch wide aluminum bars cut to

final length 30 cm. Each bar weighs 754.0 ± 0.5 g (95% CI). To track the pendulums’

positions, we illuminated the central axle with a green laser, and installed a red and a blue

LED on the top and bottom pendulum, respectively. We built a mechanical release system

to drop the pendulum from the same position in each trial (see Fig.(1)). Using a Lagrangian

mechanical model that includes the masses of the bars, bearings, and axles, we predicted

the two small-angle normal frequencies of this setup to be ω1 = 4.841± 0.042 rad/s(95%CI)

and ω2 = 12.949 ± 0.005 rad/s(95%CI).


The major difficulty of tracking multiple particles in an image sequence automatically is

to separate signals from individual particles. This was simplified by choosing red, blue, and

green for the three color indicators. Digital images are saved as a list of (R,G,B) values,

corresponding to the intensity of the red, green, and blue color channel, respectively. For a

pixel imaged from the red LED, its R value should be the maximum among the three. The

same reasoning applies to green and blue indicators as well. Therefore, this criteria allows

us to separate signals from the three different color indicators.

Rn+2 n

Rn+ n

Rn

nn

Initial position t

b

FIG. 1. We built a mechanical release system to drop the pendulum at the initial position outlined

above. The coordinate system used in our model is shown on the top right corner. The angular

position θt and θb of the the pendulums are both defined as the angle the pendulum makes with

the direction of gravity. Movies were taken at 300 fps. Three actual frames at ∆n = 60 frames

apart are shown with the schematic of the double pendulum superimposed upon them. ~Rn is the

orientation vector of the arm in the nth frame, ~Rn+∆n is this vector in the nth frame afterwards,

and ∆θ∆n is the change in angle between the two frames.

After the raw signal was decomposed for each color, we found the weighed center of inten-

sity for each color indicator. Signals below 0.2 were dropped to eliminate ripple structures


around the main peak as these ripple structures could shift the image center by about 5

pixels. Using the coordinates data we generated the orientation vector for each pendulum

in each image. For example, in the coordinate system shown in Fig.(1), the pendulum’s

orientation vector in the nth frame is ~Rn, and this vector in the next frame is ~Rn+1, and

the angular displacement is ∆θ1. The new angular position is thus θn+1 = θn + ∆θ1, where

θn is the arm’s angular position in the nth frame. For the initial configuration shown in

Fig.(1), the upper limit of ∆θ1 is about 8.5. Thus the magnitude of ∆θ1 can be found using

the principal branch of the inverse cosine function. The direction of rotation is determined

by the direction of ~Rn × ~Rn+1: if it is in the +z direction (out of the paper), we subtract

∆θ from the previous angle; if it is in the −z direction (into the paper), we add ∆θ to the

previous angle.

We verified our analysis algorithm by measuring small-angle normal frequencies ω1 and ω2

of the double-pendulum. The angular data of both pendulums were fitted to a linear combi-

nation of two damped sinusoids. We averaged the two normal frequency values obtained by

fitting the two pendulums’ angular data. The results are ω1 = 4.841 ± 0.042 rad/s(95%CI)

and ω2 = 12.949 ± 0.005 rad/s(95%CI). The predictions for ω1 and ω2 using Lagrangian

mechanics are ω1p = 4.862 ± 0.016 rad/s(95%CI) and ω2p = 13.029 ± 0.044 rad/s(95%CI).

ω1 agrees with prediction within 95% error, but ω2 does not. We plan to take more mea-

surements to investigate this discrepancy.

Now that the high-speed movie has been converted into angular positions of the two

pendulums at known times, phase plots can be made using the angular data. However,

what we are really after is the Poincare section. Following the work by Rafat, we made the

Poincare section of our double pendulum by plotting a point (θb(tc), θb(tc)) at every tc when

θt(tc) = 0 and θt(tc) > 0 [5]. In other words, we took a point from the bottom pendulum’s

phase space whenever the top pendulum crossed the equilibrium position in the positive

direction. These crossing points are highlighted in Fig.(2). To implement this algorithm,

the discrete angular data were interpolated into continuous functions θt(t) and θb(t). The tc’s

were found by solving θt(t) = 0 around each time when θt turns from negative to positive.

Typical phase plot of the bottom pendulum from a single video is shown Fig.3(b). The

pendulum’s energy dissipates due to friction. After about 20 s, it exhibits quasiperiodic large-

angle oscillations, which corresponds to the dense red region around θb = −2π. The shape

of the quasiperiodic region agrees qualitatively with the theoretical predictions. Poincare


Π2

0

Π2

Θ tra

d

5 6 7 8 9 10

4 Π8 Π

12 Π16 Π20 Π

t s

Θ bra

d

FIG. 2. 5 s of the two pendulums’ motion. Poincare section points were taken from the bottom

pendulum’s phase plots at times when the top pendulum crosses 0 in the forward direction. The

crossing points are highlighted in the graph above.

section made from 31 measurements of is shown in Fig.(4). There appears to be a periodicity

of 2π among the high angular speed points. At this point we are not sure whether this is a

peculiarity of our initial configuration or not. In the future the Poincare section of our setup

needs to be calculated. Once this peculiarity is solved, we will be ready to take more data

to construct a fractal Poincare section and measure the Lyapunov exponents of this system

quantitatively.

[1] R. DeSerio. ”Chaotic pendulum: The Complete Attractor,” Am. J. Phys. 71, 250-258 (2003).

[2] G. R. Fowles and G. L. Cassiday. ”Analytical Mechanics,” 7th edition. Cengage Learning,

(2004).


(a)

(b)

5 Π 4 Π 3 Π 2 Π Π 0 Π 2 Π 3 Π 4 Π 5 Π

20

10

0

10

20

Θb rad

Ωbra

ds

FIG. 3. (a) 60s-time-elapsed photograph of the double pendulum in motion. (b) A typical phase

plot of the bottom pendulum. The dot at (π,0) marks the starting point. The color of the plot

progresses from purple to red as time elapses. As energy damps out, the pendulum exhibits

quasiperiodic large-angle oscillations, which corresponds to the dense red region around θb = −2π.

[3] R. C. Hilborn. “Sea gulls, butterflies, and grasshoppers: A brief history of the butterfly effect

in nonlinear dynamics,” Am. J. Phys. 72, 425-427 (2004).

[4] R.B. Levien, and S. M. Tan. “Double Pendulum: Experiment in Chaos,” Am. J. Phys. 61,

1038-1044 (1993).


25 Π 20 Π 15 Π 10 Π 5 Π 0 5 Π 10 Π 15 Π 20 Π 25 Π

8 Π

4 Π

0

4 Π

8 Π

Θb rad

Ωbra

ds

FIG. 4. Poincare section of the double pendulum made from 31 measurements of its chaotic motion.

There appears to be a periodicity of 2π among the points with relatively high angular speed. At

this point we are not sure whether this is the peculiarity of our initial configuration or not. We

need to calculate the Poincare section of our setup in the future.

[5] M. Z. Rafat, M. S. Whealand, T. R. Bedding. “Dynamics of a Double Pendulum with Dis-

tributed Mass,” Am. J. Phys. 77, 3 (2009).

[6] T. Shinbrot, C. Grebogi, J. Wisdom, J. Yorke. “Chaos in a Double Pendulum,” Am. J. Phys.

60, 491-499 (1992).

Chaos of the Double Pendulum - Wabash College: Academics Physics … · · 2011-12-16WJP, PHY381...

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