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COMPLEXITY AND DYNAMIC BEHAVIOUR:
THE DOUBLE PENDULUM SIMULATION
Extracted from McLucas, A.C., 2005, ‘System Dynamics Applications: A Modular Approach to
Modelling Complex World Behaviour’, Argos, Press, Canberra, Australia
Complexity Dynamic Behaviour
In dynamic problems, the structure does not need to be constructed from complicated or
highly non-linear elements to produce complex behaviour. Complex behaviour patterns can
be produced by connections of elements or component parts that individually are very simple.
How and why this occurs is explained and demonstrated.
A Physical Example – The Simple Pendulum
By way of introduction to such complex behaviour, consider a simple physical problem. A
pendulum A-B, having a link of length l1, suspended at A, with a mass m1 located at B, is
allowed to swing freely. The link A-B is assumed to be very light (ideally having no mass)
and rigid. This pendulum is illustrated at Figure 1-1.
A
1 l1
B
m1
Figure 1-1. A Simple Pendulum
Displacement
The time-dependent displacement of mass m1 at B would be as shown in Figure 1-2.
Undamped motion of this pendulum is simple harmonic motion, following a sinusoidal
pattern shown.
t
Figure 1-2. Displacement of Pendulum Mass at B Varying with Time
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005
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Even in the case of the simple pendulum, it is the continual swapping of energy between
potential and kinetic forms, with the total energy being constant, that is of particular interest.
The swapping of energy forms occurs predictably because it is directly related to (angular)
displacement of the pendulum. See Figure 1-3. In this example, it is assumed that the lowest
point of the pendulum’s swing is taken as the datum (zero-valued level) for calculating
potential energy.
Potential energy (stored energy) is greatest when the pendulum is at the highest point of the
swing, either left or right. At this point the pendulum is stationary for an instant, and kinetic
energy (the energy of motion) is zero. At the bottom of the swing, the pendulum is moving
fastest and kinetic energy is greatest. At this point potential energy is least. The form of
energy, either potential or kinetic, follows a sinusoidal relationship over time and the graphs
are 180◦ out of phase. The total energy remains constant. This assumes, of course, that there
are no losses of energy through wind resistance or other forms of resistance to motion.
Energy
Total Energy
Kinetic
Energy
Potential
Energy
t
Figure 1-3. Energy Levels Over Time - Simple Pendulum
Extending the Physical Example - Simple Pendulum Becomes a Double Pendulum
We now modify our pendulum by attaching at B a second pendulum B-C with mass m2 at C.
This is shown diagrammatically at Figure 1-4. As A-B swings, because the position of A is
fixed, the end B is constrained to describe a circular arc. C can swing in a circular motion
around B. But B is able to move back and forth: it does not follow simple harmonic motion
as the simple pendulum A-B did.
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005
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A
1
l1
m1
B
l2
2
m2
C
Figure 1-4. A Double Pendulum System
Whilst the motion of a simple pendulum is highly predictable, the motion of the double
pendulum, with m1 at B, and m2 at C, is complex and can become chaotic.
A system or problem is described as being chaotic when it exhibits behaviour that is complex
but that complex behaviour has attributes of both randomness and apparent order, such as
patterns of behaviour over time appearing to be similar, but do not repeat exactly. Chaotic
behaviour is observed in systems or systemic problems, where small changes in initial
conditions can produce markedly different behaviour.
Assume that these pendulums can only swing in a single plane (we might visualize this plane
as the page upon which the diagram is printed, with the page held vertically). For the sake of
simplicity, we will again ignore the effects of damping (which has the effect of progressively
sapping the system’s energy) produced by resistance to motion through the air or by friction
at the joints A and B.
The algebraic expression we must formulate to describe the transferring of energy between
component parts and between the potential and kinetic forms, as facilitated by the pin linking
the two pendulums, is complicated1. To demonstrate the consequences of highly non-linear
feedback, from link A-B to link B-C and vice versa, consider Figure 1-5. This shows the
angular displacement θ1, of B relative to A, over time.
1
The transferring of energy between the links is defined in terms of angular velocity and angular
accelerations which, in turn, are dependent upon the initial angular displacements of each of the
pendulums, their masses and their lengths. Simply to demonstrate this point, instantaneous angular
accelerations for each of the links A-B and B-C respectively, are:
1 
m2l11 sin  cos   m2 g sin  2 cos   m2l222 sin   (m1  m2 ) g sin 1
2  
(m1  m2 )l1  m2l1 cos 2 
and
m2l22 sin  cos   (m1  m2 )( g sin1 cos 
 sin   g sin 2 )
 l1 12
(m1  m2 )l 2  m2 l 2 cos 
2
The equations for
instantaneous angular velocities of each of the links are similarly complicated. Energy transfer occurs
instantaneously as the combined effect of displacement, angular velocity and angular acceleration. So,
the transfers between forms of energy and between pendulums are complicated. They are highly nonlinear and their behaviour over time is impossible to describe without aid of a computer model.
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005
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1
radians
1
2
0
4
6
t
8
seconds
-1
Figure 1-5. Angular Displacement of θ1 Over Time
Figure 1-6 shows representative angular displacement θ2, of C relative to B, over time.
2
radians
12.5
10
7.5
5.0
2.5
0
2
4
6
8
t
seconds
- 2.5
Figure 1-6. Angular Displacement of θ2 Over Time
When the two angular displacements, θ1 and θ2 are plotted on the axes of the same graph, we
observe motion that is vastly different to that of a simple pendulum2. A more complete
picture of the complex nature of the swinging of the parts of our double pendulum over time
is shown at Figure 1-7.
2
Figures 1-5, 1-6 and 1-7 depict outputs of the same model. Figure 1-8 is the output of a different
model.
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005
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 2 radians
12.5
10
7.5
5
2.5
-1
1
1
radians
-2.5
Figure 1-7. Angular Displacement θ1 vs. θ2
To read this graph, start at the origin and trace along the line to its free end. At regular
intervals, stop and read the values of angular displacement from the relevant axes. The
purpose of this example is to demonstrate that dynamic behaviour (behaviour over time) can
be complex or even chaotic despite the system being made up of very simple component
parts.
The swapping of forms of energy between each of the masses is depicted at Figure 1-8. Note
that the Total Energy remains constant.
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005
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Energy
Total Energy
Kinetic Energy
Mass 2
Kinetic Energy
Mass 1
Potential
Energy
Mass 2
Potential Energy
Mass 1
t
Note: At any point in time, the addition of each of the separate energy
graphs is a constant, that is, Total Energy.
Figure 1-8. Potential and Kinetic Energy of Double Pendulum and Relationship to Total Energy
It can be daunting for us to visualize or predict the modes of swinging motion of our double
pendulum.
The single pendulum is simple. It is a second-order3 system with simple relationships
governing the flows to and from the state variables Kinetic Energy Mass1 and Potential
Energy Mass 1. We can use intuition to predict its rhythmic, stable and repeating behaviour.
Each time we start a simple pendulum swinging, we expect that its behaviour will be exactly
the same as each previous time.
We cannot use intuition to predict the behaviour of the double pendulum. Slight differences
in the initial or starting conditions can produce vastly different results.
The double pendulum problem is a fourth-order problem. That is, four state variables
describe the system. Figure 1-8 indicates this: the four state variables are Kinetic Energy
3
In this context order relates to the number of state variables which we use to describe the system.
There are two state variables, Kinetic Energy Mass1 and Potential Energy Mass1. In another context,
which you will encounter when building system dynamics computational models, order describes the
power of the terms in a polynomial expression (for example, a polynomial containing x3 terms is third
order). Such polynomial expressions are used by the system dynamics modelling software to make
numerical approximations when calculating integrations (the area under the curve describing a timedependent variable).
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005
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Mass1, Potential Energy Mass 1, Kinetic Energy Mass2 and Potential Energy Mass 2. But
the fact that this is a fourth-order problem is only a partial explanation of the difficulties we
have understanding it. What makes the behaviour of the double pendulum exceedingly
difficult to predict is that the physical feedback operates via the connecting joint at B,
instantaneously transferring energy from one pendulum to the other, is highly non-linear. The
relationships governing the transfer of energy between Kinetic Energy Mass1, Potential
Energy Mass 1, Kinetic Energy Mass2 and Potential Energy Mass 2 are complex.
Our cognitive capacity is exceeded by the co-existence of two confounding phenomenon: one
is feedback and the other is that the feedback mechanism is highly non-linear (Diehl and
Sterman, 1995; Forrester, 1987; Kleinmuntz, 1985; 1993; McLucas, 2001; 2003; Mosekilde
and Larson, 1988; Richardson, 1991; Sterman, 1989a; 1989b; 1989c.). Further, it is a harsh
reality that almost all real-world systemic problems we might wish to address are much more
complicated than a double pendulum. Forrester (1975: 63) observes that even the simplest
social systems of practical interest lie in the range of tenth to hundredth order, where order is
taken to mean the number of integrations within the interconnected feedback loops. Order
corresponds to the number of state variables in the problem of interest.
When dealing with complexity, the challenges that confront us are:

Being able to analyse problems in ways that are effective in that they extend our
comparatively limited cognitive skills. Our cognitive skills are outmatched by the
complexity of real-world problems we have to face. Kline (1995:49-68) defines a
Complexity Index, a quantitative measure, to help us categorise such problems. Kline
takes Forrester’s point about problem complexity much further and estimates that
many socio-technical problems (considered in their entirety) can be categorised by a
complexity index, C >>109. Like Kline, Diehl and Sterman (1995), Klienmuntz
(1985), Richardson (1991) and Sterman (1989) amongst others argue that our
capacity to understand feedback is poor. Sterman (2000) observes that our cognitive
capability is limited to solving reliably a first-order linear positive feedback system,
the latter being categorised by a value of C approximately equal to four (only four not
104) 4.

Being efficient. That is, analysis does not consume undue time and effort. We seek
to have methods of analysis which enhance our understanding and enables
development of clear strategies for managing complex systemic behaviours.

Learning from our analysis – we face the questions: when is this learning valuable,
and how do we know that it is?
4
For further discussion regarding the limitations of our cognitive capability in analysing and
predicting dynamic behaviour, see McLucas (2001; 2003: 28).
Prepared by: Dr Alan McLucas
Correct at: 13 September 2005