Chapter 7
Dynamical Systems
7.1
Introduction
Cellular automata, which we encountered in the previous chapter, are examples of
dynamical systems. In the present chapter, we will explore two additional dynamical systems that are not only very famous, but also nicely illustrate one fascinating
property of such systems: even very simple nonlinear dynamical systems can exhibit, despite being completely deterministic, a completely unpredictable behavior
that seem to be random – a behavior called chaos.
7.2
Definition
A dynamical system is described by the state x(t) of the system at time t, as well
as by an evolution function Φ that describes the time evolution of the system, i.e.
how the state of the system changes with time.
The state x(t) usually consists of multiple components describing different
variables of the system.
Map
When the time is discrete, the evolution function is usually called a map. It is also
not uncommon to place the time index t as a subscript of the state x to indicate
that the time is discrete.
xt+1 = Φ(xt )
(7.1)
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CHAPTER 7. DYNAMICAL SYSTEMS
This equation is sometimes also written as a difference equation:
∆xt = Φ(xt )
where xt+1 = xt + ∆xt .
Flow
When the time is continuous, the evolution function is usually called a flow. The
dynamical system is thus described by
ẋ(t) = Φ(x(t))
(7.2)
where ẋ(t) = dtd x(t) is the time derivative of the system’s state x(t) at time t.
Note that Equation 7.2 has usually multiple components, and thus constitute a
whole set of differential equations.
Non-Linearity
A dynamical system is said to be linear when the evolution function is linear, i.e.
when Φ satisfies the two following properties:
1. Additivity: Φ(x + y) = Φ(x) + Φ(y)
2. Homogeneity: λ · Φ(x) = Φ(λ · x) with λ ∈ R.
Linear systems are obviously much more convenient for mathematical analysis. Often, non-linear systems are approximated by linear systems (they are linearized), assuming very small variations of the state of the system.
We are however interested in the behavior of a less restricted set of systems
that might somehow better model phenomena observed in the “real” world. We
will thus focus on non-linear systems.
7.3
The Logistic Map
Let us consider a very simple model of population growth. The size of the population at year t is denote with xt , a number between 0 (no individual) and 1
(maximum number of individuals). We assume the following properties:
7.3. THE LOGISTIC MAP
7-3
1. (Reproduction) For small population, we assume that the population will
increase at a rate proportional to the current population. In other words, the
population at year t + 1 is proportional to the population at year t.
2. (Starvation) As the population grows towards 1 (i.e. starts reaching the maximum number of individuals), the effects of limiting factors such as diseases
or finite food supply are felt. The population will decrease at a rate proportional to the maximum population less the current population, i.e. 1 − xt .
The evolution of the population can be written as the following map:
(7.3)
xt+1 = r xt (1 − xt )
where r > 0 is a constant proportionality factor representing the combined rate
for reproduction and starvation. Figure 7.1 illustrates, for a particular value of r,
how the population varies from one year to the next.
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0
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Figure 7.1: Phase plot representing the evolution of the population, i.e. next year’s
population xt+1 as a function of the current population xt , according to Equation 7.3 (with here r = 2). The dashed lines represent the reproduction and starvation limits.
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CHAPTER 7. DYNAMICAL SYSTEMS
We will now explore the behavior of the system for different values of the
proportionality factor 0 ≤ r ≤ 4. (It can easily be shown that for r > 4, the
population can exceed the maximum value of xt = 1).
An interesting property of the logistic map (and of many other dynamical systems) is that the quantitative behavior does not depend on the particular initial
condition (in our case, the population at the initial year, x0 ) – as long as we don’t
fall into marginal conditions (in our case, a zero population, which will always
stays to xt = 0).
We thus choose to investigate, in the following sections, the behavior of the
system with an arbitrary initial population of x0 = 0.2.
7.3.1
Point Attractor
Let us consider the evolution of the population with r = 2:
x0
0.2
x1
0.32
x2
0.435
x3
0.492
x4
0.499
...
...
These values are represented graphically in Figure 7.2(a). Note that the phase
plot provides a convenient way of geometrically determining the evolution of the
population. Figure 7.2(b) illustrates the construction of the successive values of
xt using the dotted line xt+1 = xt .
For r = 2, we see that the population converges towards the value xt = 0.5.
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Figure 7.2: Evolution of the population for r = 2. (a) Population as function of
time. (b) Corresponding phase plot.
7.3. THE LOGISTIC MAP
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Figure 7.3: Evolution of the population for r = 2.7. (a) Population as function of
time. (b) Corresponding phase plot.
For r = 2.7, we see that the system converges to x ≈ 0.63 (see Figure 7.3):
x0
0.2
x1
0.432
x2
0.663
x3
0.604
x4
0.646
x5
0.618
x6
0.637
x7
0.624
...
...
For small values of r (actually, for r < 1), the populations dies out and converge towards x = 0. Figure 7.4 and the following table demonstrates this for
r = 0.5:
x0
0.2
x1
0.08
x2
0.037
x3
0.018
x4
0.009
x5
0.004
x6
0.002
x7
0.001
...
...
In summary, as long as r < 3, the population is observed to converge to a
single point in the phase plot – a point attractor.
7.3.2
Periodic Attractor
As the value of r is increased beyond 3, we observe that the population does not
converge to a single limit anymore, but starts oscillating between two different
values. Such an attractor is called a periodic attractor.
Figure 7.5 illustrates how the population of the system converges towards such
a 2-point periodic attractor for r = 3.1.
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CHAPTER 7. DYNAMICAL SYSTEMS
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Figure 7.4: Evolution of the population for r = 0.5. (a) Population as function of
time. (b) Corresponding phase plot.
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(a)
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(b)
Figure 7.5: Evolution of the population for r = 3.1. (a) Population as function of
time. (b) Corresponding phase plot.
7.3. THE LOGISTIC MAP
7-7
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(b)
Figure 7.6: Evolution of the population for r = 3.6. (a) Population as function of
time. (b) Corresponding phase plot.
7.3.3
Strange Attractor
As r increases further, the population oscillates between 4 values (for 3 < r <
3.45), 8 values (for 3.45 < r < 3.54), 16 values and so on, successively doubling
the period as r comes closer to the approximate value of 3.57.
At r ≈ 3.57 is the onset of chaotic behavior. For most values of r > 3.57, the
population doesn’t converge anymore, but has each year a different value (within
some interval). Figure 7.6 illustrates the chaotic evolution of the system’s population for r = 3.6.
Such behavior is referred to as a system with a strange or chaotic attractor.
7.3.4 Bifurcation Diagram
The various behaviors described so far can be summarized with a plot showing,
for all values of the parameters r, the limit values of the system’s population. Such
a plot, called bifurcation diagram, is shown in Figure 7.7.
The similarity of behaviors observed with the logistic map – a simple system
consisting of just one real number changing according to Equation 7.3 – is astonishingly similar to what we have encountered with cellular automata – systems
consisting of many finite state machines, all following a particular update rule.
On the one hand, both systems are observed to display point attractors, periodic attractors as well as chaotic attractors. Furthermore, there is a definite similarity between the proportionality constant r of the logistic map and Langton’s
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1
x
0
0
r
4
Figure 7.7: Bifurcation diagram of the logistic map. For each value of r, the
diagram indicates the limit population values.
λ parameter of cellular automata: as the parameter increases, the attractor of the
system changes from point to periodic to chaotic. Note however that there is no
“complex” behavior of the logistic map.
7.3.5
Fractals
There is nevertheless something interesting happening at the “edge of chaos” in
the logistic map. Figure 7.8 shows a close-up of the bifurcation diagram for approximately 2.8 < r < 4.0.
We see not only that there are some “islands of stability” within the chaotic
region, but also that the diagram shows a self-similar, fractal pattern!
Even though fractals are encountered in totally different representations (in the
patterns produced in time by cellular automata vs. patterns found in the bifurcation
diagram of the map of a dynamical system), there seems to be a profound relation
between chaos and fractals.
7.3. THE LOGISTIC MAP
Figure 7.8: Close-up of the bifurcation diagram of the logistic map.
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7.3.6
CHAPTER 7. DYNAMICAL SYSTEMS
Sensitivity to Initial Conditions
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A prime characteristic of chaotic behavior is the sensitivity to initial conditions.
Figure 7.9 illustrates how in the “ordered” regimes, a slight variation in the initial
population value is quickly “absorbed”. On the other hand, Figure 7.10 shows that
in the chaotic regime, a slight variation in the initial condition yields dramatically
different results over time.
(a) r = 2.7
(b) r = 3.1
(a) r = 3.6, 0 ≤ t ≤ 30
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Figure 7.9: The evolution of the population is observed for two slightly different
initial conditions, x0 = 0.2 (circles) and x0 = 0.22 (crosses).
(b) r = 3.6, 0 ≤ t ≤ 100
Figure 7.10: Sensitivity to initial conditions in the chaotic regime. The evolution
of the population for two slightly different initial conditions, x0 = 0.2 (circles)
and x0 = 0.22 (crosses), yield completely different results over time.
7.4. THE LORENZ ATTRACTOR
7.4
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The Lorenz Attractor
The Lorenz attractor was first studied by the meteorologist Edward Lorenz in
1963. It was derived from a simplified model of convection in the earth’s atmosphere. In this model, the atmosphere is modeled as a system with simply three
parameters coupled together through 3 non-linear differential equations.
Assigning the letters x, y and z to these three parameters, their time evolution, according to Lorenz’s model, can be expressed by the following dynamical
system:
ẋ = ay − ax
ẏ = bx − y − xz
ż = xy − cz
where a, b and c are constants. Clearly, the flow of this dynamical system (see
Equation 7.2) is non-linear because of the terms xz and xy.
Lorenz was using a computer to run his simulation of the weather. For some
reason he interrupted one of his simulations. He then wanted to see a sequence of
data again and to save time he started the simulation in the middle of its course. He
was able to do this by entering data from a printout corresponding to conditions
in the middle of his simulation which he had previously calculated.
Figure 7.11: Two different trajectories of the Lorenz simulation. There is only a
tiny difference at the starting point. This difference is sufficient for the system to
behave completely differently after some time.
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To his surprise, the weather that the machine began to predict was completely
different from the weather calculated before (see Figure 7.11). Lorenz tracked this
down to only bothering to enter 3-digit numbers into the simulation, whereas the
computer had last time worked with 5-digit numbers. This difference seemed tiny
and the consensus at the time would have been that it should have had practically
no effect. However, Lorenz had discovered that small changes in initial conditions
produced large changes in the long-term outcome.
Lorenz’s simulation is an example of a chaotic system. This illustrates once
again that chaotic systems are characterized by their extreme sensitivity to initial
conditions. If the initial state of the system is slightly different from a previous
situation, the simulation might end up producing a completely different behavior.
That is what Lorenz stumbled upon experimenting with his weather equations.
Figure 7.12: A trajectory in the Lorenz attractor.
7.4.1
The Butterfly Effect
The notion of sensitivity to initial conditions – here in the context of weather simulation, together with the typical shape of the trajectories obtained with this model
(see Figure 7.12) – is also known as the “butterfly effect”. This phrase refers to
7.5. CHAPTER SUMMARY
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the idea that a butterfly’s wings might create tiny changes in the atmosphere that
ultimately cause a tornado to appear (or prevent a tornado from appearing). The
flapping wing represents a small change in the initial condition of the system,
which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly
different.
7.5
Chapter Summary
• A dynamical system describes the time evolution of the state of a system.
• The evolution function of a dynamical system is called map when the time
is discrete, and flow when the time is continuous.
• The logistic map illustrates how a very simple, yet non-linear map can display chaotic behavior. It is a simple model of population growth.
• The behavior of population in the logistic map can reveal point, periodic or
strange attractors. Furthermore, it illustrates how chaos and fractals can be
closely related.
• The Lorenz attractor is an example of time-continuous dynamical system
capable of displaying chaotic behavior.
• The butterfly effect encapsulates the notion of sensitive dependence on initial conditions in chaos theory.
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