This document describes a study of the chaotic behavior of a magnetic pendulum. It begins by defining relevant terms like chaos and magnetic pendulum. It then derives the initial value problem that approximates the motion of the magnetic pendulum based on explicit assumptions. This derives equations for the x, y, and z components of the pendulum's motion. It shows that the system exhibits two types of chaotic behavior through numerical solutions with varying parameter values presented in graphs.

1. Chaotic Behaviour of the Magnetic Pendulum
Christopher J. Lang
20557510
December 4, 2015
Abstract
Magnetic pendulums are a simple, real-world example of chaos. Many other studies
examine the chaotic behaviour of magnetic pendulums; however, this report extends
the discussion by considering the effects of differing levels of attraction and repulsion
on this chaotic behaviour. This paper first rigorously derives the initial value problem
that approximately describes the motion of a magnetic pendulum, based on explicit
assumptions made about the model. This paper then discusses the two-fold chaotic
behaviour of that motion.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Numeric Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Appendix: Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1

2. 1 Introduction
The following subsections outline the terminology, assumptions, and symbols used in the
report.
1.1 Terminology
The following terms are used throughout this report:
1: Chaos is a difficult concept to define mathematically. Many chaos scientists and text-
books cannot agree on a single definition of chaos [4]. In this report we use Lorenz’s
definition:
Chaos: When the present determines the future, but the approximate
present does not approximately determine the future [1].
In other words, chaos is the sensitivity of a system to initial conditions.
2: A magnetic pendulum is quite similar to the simple pendulum; however, there are a
few alterations. In this paper, we use the following definition:
Magnetic Pendulum: A pendulum with a magnet acting as a bob, which
is attracted or repelled by magnets lying on a plane unreachable by the
pendulum .
1.2 Assumptions
The following assumptions are used in this model:
1: The observations are made in an inertial reference frame.
2: The pendulum is centred on the origin of the xy-plane of a Cartesian coordinate system,
where height is measured upwards.
3: The pendulum is anchored such that the pendulum lies above the xy-plane, the plane
upon which the other magnets are placed.
4: The length of the pendulum is constant.
5: The minimum height of the bob is much less than the length of the pendulum.
6: The magnets are point attractors.
7: The spacing of the magnets is much smaller than the length of the pendulum.
8: The pendulum does not travel far from the magnets.
9: The pendulum starts from rest.
2

3. 10: The magnitude of the magnetic force between two point attractors is inversely pro-
portional to the square of the distance between them. Furthermore, it points in the
direction of the relative position of the magnet with respect to the bob.
11: The drag force that acts on the pendulum bob is linearly proportional to its velocity
and opposes motion.
1.3 Symbols
The following table outlines the various symbols used throughout the report. Note that
a boldface symbol is a vector. Also, a dot above a symbol indicates taking the derivative
with respect to time. Furthermore, a symbol between { and } indicates a set, whose upper
and lower bounds are represented by superscripts and upperscripts after the }, respectively.
Finally, note that the table is in alphabetical order, with greek letters appearing alphabeti-
cally after latin letters.
Symbol Description
b
Proportionality constant for
drag force.
F Net force on the pendulum.
Fd
Drag force on the pendulum
bob.
Fg
Force of gravity on the pendu-
lum bob.
{Fmi
}n
i=1
Magnetic force on the pendulum
bob by each magnet.
g
The magnitude of the accelera-
tion due to gravity.
h
The minimum height of the pen-
dulum above the plane.
i
Index for the magnets on the
plane.
ˆk Unit vector for the z-axis.
{ki}n
i=1
Proportionality constant for
each magnetic force.
L The length of the pendulum.
m Mass of the pendulum bob.
n
The number of magnets on the
plane.
P The ratio of b to m.
Symbol Description
Q The ratio of g to L.
r Position of the pendulum bob.
{Ri}n
i=1 The ratio of each ki to m.
T
Tension force on the pendulum
bob.
Tx
x-coordinate of the tension
force.
Ty y-coordinate of the tension force.
Tz z-coordinate of the tension force.
x
x-coordinate of the position of
the pendulum bob.
x0
x-coordinate of the initial posi-
tion.
y
y-coordinate of the position of
the pendulum bob.
y0
y-coordinate of the initial posi-
tion.
z
z-coordinate of the position of
the pendulum bob.
θ
Azimuthal angle of the tension
force.
φ
Inclination angle of the tension
force.
3

4. 2 Equations
In this section we will derive and review some of the equations that will be used in de-
riving the initial value problem.
As we are in an intertial reference frame (assumption 1), from Newton’s Second Law we
have:
F = m¨r
As the forces that act on the pendulum bob are: Fg, Fd, T, and {Fmi
}n
i=1,
Fg + Fd + T +
n
i=1
Fmi
= m¨r (2.1)
From assumption 11 we have that Fd ∝ ˙r. As the drag force opposes motion, let b ∈ R,
b > 0, such that,
Fd = −b˙r (2.2)
Let i ∈ N, i ≤ n. Also, let ri := ri − r. As we are dealing with point attractors,
assumption 6, we have that Fmi
∝ 1
ri
2 ˆri, from assumption 10. Let ki ∈ R such that,
Fmi
=
ki
ri
2ˆri
As ˆri =
ri
ri
,
Fmi
=
ki
ri
3 ri (2.3)
3 Analysis
In this section we will derive the simplified initial value problem that approximately de-
scribes the behaviour of the magnetic pendulum, given our assumptions.
From assumption 4 we have that L is constant. Let h be the minimum height of the
pendulum. As the pendulum is centred on the origin of the xy-plane of a Cartesian coordinate
system, assumption 2, for a given r = (x, y, z) we have that the position of the bob, relative
to the anchor, is: (x, y, z − L − h). As the length of the pendulum is L,
x2
+ y2
+ (z − L − h)2
= L2
Solving for z,
z = L + h ± L2 − x2 − y2
4

5. As the pendulum bob is close to the magnets, assumption 8, we must reject the plus
option in the above ±. Thus,
z = L + h − L2 − x2 − y2 (3.1)
From assumption 8 we have that the pendulum does not travel far from the magnets.
Furthermore, from assumption 7 we have that the spacing of the magnets is much less that
length of the pendulum. Thus, x and y are much smaller than L.
Expanding L2 − x2 − y2 using the Maclaurin series,
L2 − x2 − y2 =
√
L2 − 02 − 02 −
x2
2
√
L2 − 02 − 02
−
y2
2
√
L2 − 02 − 02
+ . . .
As x and y are small compared with L, we ignore the quadratic and higher degree terms,
leaving:
L2 − x2 − y2 ≈ L
Therefore,
z ≈ h (3.2)
Thus, ˙z ≈ 0 and ¨z ≈ 0, as z is approximately constant.
As Fg = −mgˆk, consider the z-component of (2.1), substituting (2.2) and (2.3),
−mg − b ˙z + Tz +
n
i=1
ki
(xi − x)2 + (yi − y)2 + (zi − z)2
3 (zi − z) = m¨z
From assumption 3 we have that the magnets all lie on the xy-plane. Thus, zi = 0,
∀i ∈ N, i ≤ n. As z ≈ h, ˙z ≈ 0, and ¨z ≈ 0,
−mg + Tz − h
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 = 0
Solving for Tz,
Tz = mg + h
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3
As Tz = T cos φ, we get
T cos φ = mg + h
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 (3.3)
Solving (3.1) for L + h − z, we get
5

6. L + h − z = L2 − x2 − y2 (3.4)
As cos φ = L+h−z
L
, substituting (3.4),
cos φ =
L2 − x2 − y2
L
Substituting into (3.3) and solving for T ,
T =
mgL
L2 − x2 − y2
+
hL
L2 − x2 − y2
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 (3.5)
As Tx = − T sin φ cos θ,
Tx = − sin φ cos θ



mgL
L2 − x2 − y2
+
hL
L2 − x2 − y2
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3



(3.6)
As cos φ =
√
L2−x2−y2
L
, it follows that sin φ =
√
x2+y2
L
, as sin φ ≥ 0, as 0 ≤ φ ≤ π.
Furthermore, cos θ = x√
x2+y2
, so sin φ cos θ = x
L
. Substituting into (3.6) and simplifying,
Tx = −
mgx
L2 − x2 − y2
−
hx
L2 − x2 − y2
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3
As L2 − x2 − y2 ≈ L,
Tx = −
mgx
L
−
hx
L
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3
From assumption 5, h is small compared with L. As h and x are small compared with
L, we ignore any quadratic terms. Thus,
Tx = −
mg
L
x (3.7)
Similarly, Ty = − T sin φ sin θ. As sin θ = y√
x2+y2
and sin φ =
√
x2+y2
L
, it follows that
sin φ sin θ = y√
x2+y2
. Thus,
Tx = −
mgy
L2 − x2 − y2
−
hy
L2 − x2 − y2
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3
6

7. Using L2 − x2 − y2 ≈ L and ignoring the quadratic terms,
Ty = −
mg
L
y (3.8)
As z ≈ h, ˙z ≈ 0, and ¨z ≈ 0, we can ignore z. Substituting (2.2), (2.3), (3.7), and (3.8)
into (2.1),
−b ˙x −
mg
L
x +
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 (xi − x) = m¨x
−b ˙y −
mg
L
y +
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 (yi − y) = m¨y
Dividing by m, as m = 0, and rearranging,
¨x +
b
m
˙x +
g
L
x +
1
m
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 (x − xi) = 0
¨y +
b
m
˙y +
g
L
y +
1
m
n
i=1
ki
(xi − x)2 + (yi − y)2 + h2
3 (y − yi) = 0
Let P := b
m
, Q := g
L
, and Ri := ki
m
, ∀i ∈ N, i ≤ n. Then our system is:
¨x + P ˙x + Qx +
n
i=1
Ri
(xi − x)2 + (yi − y)2 + h2
3 (x − xi) = 0
¨y + P ˙y + Qy +
n
i=1
Ri
(xi − x)2 + (yi − y)2 + h2
3 (y − yi) = 0
From assumption 9, any solution to this system must satisfy ˙x(0) = 0 and ˙y(0) = 0.
Note that this assumption helps ensure that the system satisfies assumption 8 as it evolves.
This is because assumption 9 makes sure the pendulum cannot start by going near the speed
of light, for instance, which would certainly make the pendulum travel far away from the
magnets. Moreover, any solution to this system must also satisfy x(0) = x0 and y(0) = y0,
where x0, y0 ∈ R are small, in order for the system to satisfy assumption 8.
7

8. Summarizing, our initial value problem is:
¨x + P ˙x + Qx +
n
i=1
Ri
(xi − x)2 + (yi − y)2 + h2
3 (x − xi) = 0 (3.9)
¨y + P ˙y + Qy +
n
i=1
Ri
(xi − x)2 + (yi − y)2 + h2
3 (y − yi) = 0 (3.10)
satisfying: ˙x(0) = 0, ˙y(0) = 0, x(0) = x0, and y(0) = y0.
4 Numeric Solutions
In this section we demonstrate the two-fold chaotic behaviour of the magnetic pendulum
through graphs calculated using numeric methods.
Table 1 outlines the values of the various parameters of the system for figures 1 through
7:
Figure P Q h n {Ri}n
i=1 {xi}n
i=1 {yi}n
i=1
1 0.2 0.1 0.5 2 10, 10 1, -1 0, 0
2 0.2 0.1 0.5 2 20, 10 1, -1 0, 0
3 0.2 0.1 0.5 3 10, 10, 10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
4 0.2 0.1 0.5 3 10, 10, -10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
5 0.2 0.1 0.5 3 -10, -10, -10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
6 0.2 1.0 0.5 3 10, 10, 10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
7 2.0 1.0 0.5 3 10, 10, 10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
Table 1: Parameter values for chaotic plots
These values for the parameters were chosen to demonstrate the effects that: the number
of magnets, repulsion, differing levels of attraction, drag, and gravity can have on the chaotic
behaviour of the system.
The following figures represent the ending positions of the pendulum based on various
starting points. Any given pixel represents the initial point and is assigned a colour depend-
ing on where it tends towards or away from for high values of t.
8

9. In figure 1 there are two equally attracting magnets placed at (1, 0) and (−1, 0). It is
evident from the figure that the system is quite chaotic when the initial point is far from
the magnets; however, as the initial point approaches the magnets, this chaotic behaviour
disappears.
In figure 2 there are two attracting magnets placed at (1, 0) and (−1, 0). From table 1
we see that the magnet at (1, 0) has twice the level of attraction compared to the magnet at
(−1, 0). The figure clearly shows that, while there is still some chaos, the differing levels of
attraction has significantly reduced the chaotic behaviour of the system.
In figure 3 there are three equally attracting magnets placed at (
√
3
2
, 1
2
), (−
√
3
2
, 1
2
), and
(0, −1). This scenario is the same as in 1, except that one magnet has been added. Addi-
tionally, the magnets have been shifted to better demonstrate the symmetry of the chaos.
When comparing with figure 1, the regions of low chaotic behaviour are smaller; however,
once you leave these regions, the behaviour is immediately much more chaotic than the case
with two equally attracting magnets. This shows that increasing the number of magnets has
increased the level of chaotic behaviour in the system.
Figure 4 is essentially the same scenario as in figure 3; however, the bottom magnet repels
the pendulum bob. Observe, from table 1, that the magnitude of the repulsion is the same
as the attraction of the other two magnets. It is interesting to note that around the two
attracting magnets, we observe behaviour similar to that of the case of two equally attracting
magnets. Although, around the repelling magnet, we see much more chaotic behaviour than
for the other magnets; however, not as much chaotic behaviour as in the case of the three
equally attracting magnets.
In figure 5 we essentially have the same scenario as in figure 3; except that the magnets
are now equally repelling the pendulum, not attracting it. Due to the nature of being repelled
by all of the magnets, the meaning of the colours has changed for this figure. In this figure,
a coloured point indicates from which magnet the final position of the magnet is furthest,
instead of to which magnet it is closest. The white points in the middle indicate initial points
where the pendulum tends towards the origin for high values of t. It is interesting to observe
that the three equally repelling magnets greatly reduce the chaotic behaviour of the system,
as the plane is almost equally divided into three regions. Along with figure 4, these figures
demonstrate that adding repulsion into the system reduces the amount of chaos significantly.
Figure 6 is the same system as figure 3, but the pendulum is being influenced by a stronger
gravity field. When comparing the two figures, it is clear that they are extremely similar.
They only differ in that the regions of no chaos in figure 6 are slightly smaller than in figure
3 and that there is more chaotic behaviour in figure 6 than in figure 3. This demonstrates
the little impact that gravity has on the chaotic behaviour of the system.
In figure 7 we have the same scenario as in figure 6, but the system is influenced by
a stronger drag force. It is clear from the figure that the level of chaotic behaviour has
decreased significantly. This indicates that the drag force has a large effect on the chaotic
9

10. behaviour of the system.
Table 2 outlines the values of the various parameters of the system for figures 8 and 9:
Figure x0 y0 P Q h n {Ri}n
i=1 {xi}n
i=1 {yi}n
i=1
8 1.00 0.00 0.2 0.1 0.5 3 10, 10, 10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
9 1.00 0.05 0.2 0.1 0.5 3 10, 10, 10
√
3
2
, −
√
3
2
, 0 1
2
, 1
2
, -1
Table 2: Parameter values for pendulum paths
The only change in the parameters for these figures is the starting point of the pendulum.
This is done to demonstrate that the path taken by the pendulum is chaotic even when it
tends towards the same magnet.
Figure 8 has the pendulum starting at (1, 0), whereas figure 9 has the pendulum starting
a small distance away, (1, 0.05). When comparing the two figures, it is interesting to note
that there are much smaller oscillations along the line y = x in figure 9 than in figure 8.
This is despite a vertical translation of the starting point of only 0.05.
5 Conclusions
With an appropriate number of assumptions, we have derived the approximate system
of equations of motion for a magnetic pendulum. Furthermore, with the use of numeric
solutions, we have demonstrated the chaotic behaviour of a magnetic pendulum, not only in
the end points of the system, but in the paths taken to these end points.
Our assumptions are made so that the motion of the pendulum is small enough that the
motion of the pendulum is effectively confined to a plane elevated above the magnets. This
assumption allows us to reduce the system of three equations to two simpler equations.
We then use numeric solutions to show the two-fold chaotic behaviour of the magnetic
pendulum. This is done by constructing plots showing the endpoints of various initial values,
as well as plots that show the path taken to these endpoints.
These numeric solutions have demonstrated that the magnetic has a significant amount
of chaotic behaviour, not only in its endpoints, but its paths to these endpoints as well.
Furthermore, we have shown that: the number of magnets, drag force, differing levels of
attraction, and repulsion have a significant impact on the amount of chaotic behaviour of
the system, whereas gravity has little impact.
Due to the form of the magnetic force, assumption 10, these results could be easily ap-
plied to a charged pendulum bob that is influenced by charges lying on a plane. Additionally,
10

11. one could extend this research to consider attractors that are not point attractors and dipole
magnets. Furthermore, the assumption of the vertical motion being confined could be elimi-
nated, as the system would still in terms of two coordinates, as it is confined by the length of
the pendulum. However, additional assumptions may still be needed in order to the reduce
the equations of motion to a simpler, more feasible system.
11

12. References
[1] Danforth, Christopher M. ”Chaos in an Atmosphere Hanging on a Wall.” From
Mathematics of Planet Earth. http://mpe2013.org/2013/03/17/chaos-in-an-atmosphere-
hanging-on-a-wall/
[2] Sunde, Liam L. ”The Magnetic Pendulum.” From Northwestern Physics.
http://diablo.phys.northwestern.edu/ mvelasco/330-2/Liam NonLinear.pdf
[3] T´el, Tam´as, & Gruiz, M´arton. (2006). Chaotic Dynamics: An Introduction Based on
Classical Mechanics. Cambridge University Press.
[4] Weisstein, Eric W. ”Chaos.” From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Chaos.html
[5] Win, Ian J. ”The Chaotic Oscillating Magnetic Pendulum.” From Harvey Mudd College.
https://www.math.hmc.edu/ dyong/math164/2006/win/presentation.pdf
6 Appendix: Figures
Figure 1: This figure shows the initial points where the pendulum tends towards (1, 0) (blue)
or (−1, 0) (green) for high values of t.
12

13. Figure 2: This figure shows the initial points where the pendulum tends towards (1, 0) (blue)
or (−1, 0) (green) for high values of t.
Figure 3: This figure shows the initial points where the pendulum tends towards (
√
3
2
, 1
2
)
(blue), (−
√
3
2
, 1
2
) (green), or (0, −1) (red) for high values of t.
13

14. Figure 4: This figure shows the initial points where the pendulum tends towards (
√
3
2
, 1
2
)
(blue) or (−
√
3
2
, 1
2
) (green) for high values of t.
Figure 5: This figure shows the initial points where the pendulum tends away from (
√
3
2
, 1
2
)
(blue), (−
√
3
2
, 1
2
) (green), or (0, −1) (red) or towards the origin (white) for high values of t.
14

15. Figure 6: This figure shows the initial points where the pendulum tends towards (
√
3
2
, 1
2
)
(blue), (−
√
3
2
, 1
2
) (green), or (0, −1) (red) for high values of t.
Figure 7: This figure shows the initial points where the pendulum tends towards (
√
3
2
, 1
2
)
(blue), (−
√
3
2
, 1
2
) (green), or (0, −1) (red) for high values of t.
15

16. Figure 8: This figure shows the path taken by the pendulum when it starts at (1, 0).
Figure 9: This figure shows the path taken by the pendulum when it starts at (1, 0.05).
16