A Route to Chaos for the Physical Double Pendulum by
1. A Route to Chaos for the Physical Double
Pendulum by Examining the Case of Near
Integrability
Daniel Berkowitz
New York University School of Engineering,Department of Applied Physics
db2505@nyu.edu
2015
Abstract
The aim of this paper is to show through computational methods
that a route to chaos exists for the physical double pendulum similar to
that of the driven damped pendulum. The Lagrangian of the physical
double pendulum is given, from which the Hamiltonian is derived. This
Hamiltonian is expanded to first order in , where is the distance be-
tween the pivot point of the bottom pendulum and its center of mass.
When = 0 the physical double pendulum is integrable. Because the
case of small is treated, in this paper the physical double pendulum is
a near integrable system. Poincare sections will be created for different
initial conditions in which the physical double pendulum starts at rest.
These sections are formed by plotting the angular position and momen-
tum of the bottom pendulum when the top pendulum has an angular
velocity of zero. It will be observed that for values of just before the
onset of global chaos, the quasi periodic tori in the Poincare sections
collapse to points, and the bottom pendulum exhibits period n (i.e. n
oscillations per period) motion. The evidence for this route to chaos is
further justified from bifurcation diagrams for which is varied. Right
before global chaos, the bifurcation diagrams splits corresponding to
the period n orbits shown to exist in the Poincare sections.
1 Introduction
The physical double pendulum is a simple yet physically rich system. Easy
to construct, the analysis of its physical phenomena and the associated sub-
1
2. tleties are sophisticated and compelling. It belongs to a large class of non
linear Hamiltonian systems with two degrees of freedom. One of the virtues
of its analysis is that it sheds light on the behavior of more obscure systems
in its class.
The simple double pendulum as shown by (Fig. 1)see [2] is a simple
pendulum with a second simple pendulum attached to the bob of the first.
This is difficult to work with experimentally because the strings deform and
bend, introducing more degrees of freedom. The physical double pendu-
lum, on the other hand, as seen by (Fig.2) see [3], is constructed from rigid
materials, and can be treated both theoretically and experimentally as a
system with only two degrees of freedom. This paper demonstrates that the
physical double pendulum, despite being a simple system, exhibits many
surprising phenomena beyond that of merely being chaotic. Computational
analysis shows that its route to chaos is characterized by the tori of its phase
space manifold collapsing in on themselves as the parameter is increased.
These tori eventually collapse into points, indicating periodic motion ev-
ery n oscillations, prior to the onset of global chaos. Once this route to
chaos is described, associated questions and ideas for further research will
be discussed.
Fig.1. Simple Double Pendulum (Image taken from reference [2])
2
3. Fig.2. Physical Double Pendulum (Image taken from reference [3])
2 Deriving the Hamiltonian
L =
1
2
I1
˙θ2
1+
1
2
I2
˙θ2
2+
1
2
m2h2
1
˙θ2
1+
1
2
m2d2
2
˙θ2
2+m2h1d2
˙θ1
˙θ2cos(θ1−θ2)+m1gd1cos(θ1)+m2gh1cos(θ1)+m2gd2cos(θ2)
(1)
The Lagrangian for the physical double pendulum is given in the expres-
sion above. I1 is the moment of inertia of the top pendulum taken about its
pivot point;m1 and m2 are the masses of the top and bottom pendulums,
respectively; h1 is the length of the top pendulum; d1 is the distance between
the pivot point of the top pendulum and its center of mass; and I2 is the
moment of inertia of the bottom pendulum taken with respect to its center
of mass rather than with respect to its pivot point; The key parameter for
our purposes is d2, the distance between the pivot point of the bottom pen-
dulum and its center of mass. When d2=0, the physical double pendulum
becomes an integrable system of a simple pendulum with a rotor attached
at the bottom (see [1]). In this study, d2 is small, i.e., on the order of .
This simplifies the Hamiltonian significantly.
With this in mind, the new Lagrangian is given below.
3
4. L =
1
2
I1
˙θ2
1+
1
2
I2
˙θ2
2+
1
2
m2h2
1
˙θ2
1+
1
2
m2
2 ˙θ2
2+m2h1
˙θ1
˙θ2cos(θ1−θ2)+m1gd1cos(θ1)+m2gh1cos(θ1)+m2g cos(θ2)
(2)
To further simplify the calculation, all the parameters besides will be
set equal to unity.
L = ˙θ2
1 +
1
2
˙θ2
2 +
1
2
2 ˙θ2
2 + ˙θ1
˙θ2cos(θ1 − θ2) + 2cos(θ1) + cos(θ2) (3)
H = ˙θ1p1 + ˙θ2p2 − L (4)
pi =
∂L
∂ ˙θi
(5)
The Hamiltonian is a function of the canonical momenta pi and the
generalized coordinates θi only. With pi= ∂L
∂ ˙θi
one can solve the ˙θi in terms
pi and substitute them into (4) to get the Hamiltonian
p1 =
∂L
∂ ˙θ1
= 2 ˙θ1 + (θ1 − θ2) ˙θ2 (6)
p2 =
∂L
∂ ˙θ2
= (θ1 − θ2) ˙θ1 + ˙θ2 + 2 ˙θ2 (7)
We have two equations with two unknowns ˙θ1 and ˙θ2.Using Mathematica,
we solve these equations to get the generalized velocities in terms of the
momenta
˙θ1 =
p1 + 2
p1 − cos(θ1 − θ2)p2
2 + 2 2 − 2cos(θ1 − θ2)2
(8)
˙θ2 =
sec(θ1 − θ2)( p1 − 2p2sec(θ1 − θ2))
2 − 2sec(θ1 − θ2)2 − 2 2sec(θ1 − θ2)2
(9)
4
5. We insert this into the equation for the Hamiltonian and simplify (again
using Mathematica) to obtain
H = −2cos(θ1) − cos(θ2) +
3(1 + 2
)p2
1 − 2 cos(θ1 − θ2)p1p2 − 2p2
2
4 + 3 2 − 2cos(2(θ1 − θ2))
(10)
Now the crucial step is to expand this Hamiltonian to first order in to
obtain
H = −2cos(θ1) +
p2
1
4
+
p2
2
2
− 4 (−cos(θ2) −
1
8
cos(θ1 − θ2)p1p2) (11)
When is zero the new Hamiltonian is that of a simple pendulum with
a free rotor attached. This system is integrable because it has two degrees
of freedom and two conserved quantities, the time independent Hamiltonian
and the momentum of the free rotor p2. When is not zero the Hamiltonian
is that of an integrable system with a small non-linear perturbation, turning
it into a non integrable system. In this paper due to the smallness of the
system is referred to as being near integrable. A Hamiltonian of this form
can be written in terms of the action angle variables Ji and θi. In such
case, an analysis of the frequency can be conducted by using the formula
below, where ωi is the frequency of either the top pendulum or the bottom
pendulum.
∂H
∂Ji
= ωi (12)
Though not done in this paper, this analysis would be fruitful for certain
values of as will be shown.
5
6. 3 The Route to Chaos of the Double Pendulum
Fig.3.
The above in Fig 3 is a Poincare section when the initial momentum is
zero, and the initial angles are both set to 1.5 radians. The Y axis is the
momentum of the bottom pendulum, and the X axis is its angle in radi-
ans. The values are sampled when the velocity of the upper pendulum is
zero. The value of is .1089 meters and as can be seen the pendulum is
approximately executing period n motion. Note that the dots are discrete
and do not smear out or increase in number as a function of time. For
a physical single pendulum, if a Poincare section were created by plotting
points every time the pendulum had a velocity of zero, only one point would
be visible. This is because the system has period 1 motion. Here, the exis-
tence of multiple points indicates that the motion of the bottom pendulum
is periodic over a number n of its oscillations. This is period n motion. It is
important to note that from the above Poincare section, one cannot deduce
the number of oscillation per period. Because the points in the Poincare
section are plotted only when the top pendulum has an angular velocity of
zero, states for which the bottom pendulum has an angular velocity of zero
may be missed. This method of constructing Poincare sections was chosen
because it illuminates how quasi periodic tori form and then collapse into
themselves to form period n orbits.
This value of was found using trial and error. First, an animation of
6
7. a Poincare section for a given initial condition was run over a large range
of . Once the tori were observed to collapse into points the range of
was narrowed for a given precision. For this case the precision was set to
four decimal places. If the value of was calculated to a higher precision,
the motion would be closer to a perfect period n oscillator. However, as
the graph of θ vs t for the bottom pendulum Fig 4 shows, even at the
critical point =.1089, the motion of the bottom pendulum is virtually
indistinguishable from that of a period n oscillator.
Fig.4. Note that the graph shown in Fig.4. is taken for a time interval
of 300 seconds. The graph shows period n motion. To further give evidence
that this is not transient motion, the graph in Fig 5 covers a range of t up
to 3000 seconds.
7
8. Fig.5.
The Poincare section below shows = .1090 meters.
Fig.6.
Note that the dots are now somewhat smeared out, which indicates that
the pendulum is no longer executing period n motion. This shows that ≈
.1089 meters is a very special value. The code is provided in the appendix
at the end of this paper. Others may wish to create their own Poincare
8
9. section animation for different initial conditions to observe other values of
for which the physical double pendulum exhibits period n motion.
Fig.7.
This Poincare section Fig 7 is when is less then .1089 meters and as can
be seen the physical double pendulum is exhibiting quasi periodic motion,
and the Poincare section is a cross-section of a Cantor family invariant tori.
According to the Liouville Arnold Theorem, an integrable system has the
phase space manifold of an n-torus, where n is number of degrees of freedom
of the system. The physical double pendulum has two degrees of freedom,
and thus has a four-dimensional phase space manifold. Because of this, in
order to study its phase space manifold, Poincare sections have to be made.
For a system which can be modeled as an integrable system with a pertur-
bation for small values of the phase space manifold will be an invariant
tori. As the value of increases, the tori break down and becomes a topo-
logically complicated figure, usually manifesting itself as scattered dots all
over the Poincare section. This phenomenon of invariant tori breaking down
as increases is governed by the KolmogorovArnoldMoser theorem (KAM
theorem) see [4]. While in the quasi periodic regime of initial conditions and
parameters, the Poincare sections of these chaotic systems are tori. For the
physical double pendulum as can be seen in Fig.7 and Fig.3 the Poincare
sections start out as an invariant tori and then breaks down by collapsing
in on themselves to form period n orbits. Then once is further increased,
9
10. the collapsed tori expand rapidly becoming a sea of scattered dots.
Fig.8.
The above is a Poincare section for = .13 meters in which global chaos
has emerged. If the animation were run, one would see the period n Poincare
section getting larger and eventually expanding into the sea of points as
shown.
10
11. Fig.9.
Note that in Fig. 9 the Y axis is the angle of the bottom pendulum, and
the X axis shows ranging from 0 to .150 meters. Every time the velocity
of the top pendulum is zero, the angle of the bottom pendulum at that
specific value of is plotted. Some interesting phenomena can be inferred
from this bifurcation diagram. At around ≈ .1089, a splitting occurs that
is reminiscent of the period doubling cascade associated with the bifurcation
of a driven damped pendulum.
Further evidence is given below by Poincare sections for other initial
conditions. In these sections the values of are given in which the bottom
pendulum executes approximately period n motion. In addition to Poincare
sections, graphs of θ vs t are given, as well as bifurcation diagrams similar
to the one for the initial conditions 1.5 radians.
Fig.10.
Initial conditions of 1 radian for both of the angles at =.73625
11
15. Fig.17.
Fig.18.
As the energy of the physical double pendulum is increased, the value of
associated with period n motion decreases. This result is intuitive in that
larger values of correspond to greater torque associated with the lower pen-
dulum, and larger initial angles result in more potential energy for motion.
In all the bifurcation diagrams before global chaos emerges, a splitting oc-
curs because the invariant tori collapses at the approximate values of listed
above. For large we observe this type of splitting as well. In Fig.18, unlike
the other bifurcation diagrams in which we varied , in this one the initial
angles were varied; both of the initial angles were kept the same. Every time
the bottom pendulum has a velocity of zero, the angular position is plotted.
At approximately 0.89 radians, a split occurs in which period n motion takes
15
16. place. When the initial angles are further increased, global chaos emerges.
Fig.19.
The code for the bifurcation diagram below is given the appendix
4 Conclusion and Suggestion for Further Work
The Hamiltonian was derived for a physical double pendulum and then
truncated to first order in . Poincare sections were created for some initial
conditions, and it was observed that for certain values of , period n orbits
existed. The transition from quasi periodic to period n motion was shown in
the Poincare sections to occur because the quasi periodic tori collapse into
points as approaches a specific value. By viewing the bifurcation diagrams,
it was shown that period n orbits manifest themselves as sudden splits and
always precede the onset of global chaos. These results show that prior to
the onset of global chaos, the physical double pendulum goes through period
n motion. This is an interesting phenomenon. As the small perturbation
is initially increased the motion of the bottom pendulum is quasi periodic.
Rather than becoming chaotic as is increased even further, the motion
16
17. becomes more orderly up until a point. Instead of transitioning to chaotic
motion by starting off orderly and increasingly becoming more disorderly
as is increased, the motion becomes periodic at a certain value of . It
then quickly becomes very disorderly. This behavior of the physical double
pendulum is just an example of how chaotic systems can defy common sense.
A systematic analytical method should be developed, which can at least
approximate the values of which the physical double pendulum exhibit
period n motion.
5 References
1.Melnikov’s method applied to the double pendulum In: Zeitschrift fr
Physik B Condensed Matter. (Zeitschrift fr Physik B Condensed Matter,
December 1994, 93(4):521-528) Publication Information: Springer-Verlag
Publication Year: 1994
2.Deleanu, Dumitru1 dumitrudeleanu@yahoo.com Annals of the Univer-
sity Dunarea de Jos of Galati: Fascicle II, Mathematics, Physics, Theoret-
ical Mechanics. 2011, Vol. 34 Issue 2, p203-211. 9p. Maritime University
of Constanta, Mathematical Sciences Department, 104 Mircea cel Batran
Street, Constanta, Romania ISSN 2067-2071
3.http://psi.nbi.dk/@psi/wiki/TheR. Nielsen (23051992), E. Have (08091993),
B. T. Nielsen (15051991) Supervisors: Namiko Mitarai, Jrg Helge Mller
Project 2013-14 The Niels Bohr Institute
4.Xuezhu Lu1 xujun@seu.edu.cn Junxiang Xu1 jennysundate@gmail.com
Journal of Mathematical Physics. 2014, Vol. 55 Issue 8, p082702-1-082702-
12. 12p. Department of Mathematics, Southeast University, Nanjing 210096,
People’s Republic of China ISSN 0022-2488
6 Appendix
This is the code to animate Poincare sections for a varying value of
17
![tleties are sophisticated and compelling. It belongs to a large class of non
linear Hamiltonian systems with two degrees of freedom. One of the virtues
of its analysis is that it sheds light on the behavior of more obscure systems
in its class.
The simple double pendulum as shown by (Fig. 1)see [2] is a simple
pendulum with a second simple pendulum attached to the bob of the first.
This is difficult to work with experimentally because the strings deform and
bend, introducing more degrees of freedom. The physical double pendu-
lum, on the other hand, as seen by (Fig.2) see [3], is constructed from rigid
materials, and can be treated both theoretically and experimentally as a
system with only two degrees of freedom. This paper demonstrates that the
physical double pendulum, despite being a simple system, exhibits many
surprising phenomena beyond that of merely being chaotic. Computational
analysis shows that its route to chaos is characterized by the tori of its phase
space manifold collapsing in on themselves as the parameter is increased.
These tori eventually collapse into points, indicating periodic motion ev-
ery n oscillations, prior to the onset of global chaos. Once this route to
chaos is described, associated questions and ideas for further research will
be discussed.
Fig.1. Simple Double Pendulum (Image taken from reference [2])
2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)