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Chaos and the Logistic Map
PHYS220 2004
by Lesa Moore
DEPARTMENT OF PHYSICS
Macquarie University 1
Different Types of Growth
Arithmetic Growth y = 2x + 10
70
60
50
40
y
30
20
10
0
0 5 10 15 20 25 30
x
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Quadratic Growth
y = x 2 - 2E-13x
1200
1000
800
y(x)
600
400
200
0
0 5 10 15 20 25 30 35
x
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Cyclic Growth & Decay
1.5
1
0.5
y(x)
0
0 1 2 3 4 5 6 7
-0.5
-1
-1.5
x
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Geometric Growth 0.4055x
y=e
45
40
35
30
25
y(x)
20
15
10
5
0
0 2 4 6 8 10
x
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A Population Example
Every generation, the population of fish
in a lake grows by 10%.
Nn is the population of generation n.
r=1.1 is the constant growth rate.
The difference equation is: Nn+1=rNn.
The population sequence for N1=100 is:
100, 110, 121, 133, 146, 161, …
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The Analytical Solution
The rate of change of population N is:
dN
N
dt
dN
Separating the variables: dt
N
And integrating both sides: N
dN '
t
N ' dt '
N0 0
N
N t
ln
0
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Final Steps
Exponentiating both sides:
N t
N e
0
t
Yields: N N 0e
This example exhibits geometric growth and
the analytic solution is an exponential
function.
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These Systems are Predictable
Arithmetic, quadratic and geometric
growth, and cyclic growth and decay
are predictable systems with analytical
solutions.
The state x(t) at time t may be
predicted from the state at time t=0
using an analytical formula.
Predictable for bank loans, filling a
water tank, a “simple” pendulum.
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Linearity
Linear systems are easy to understand:
double the input yields double the
output.
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Unpredictability
Not all systems are predictable.
Some systems have no analytical
solutions.
We now consider a different type of
growth, known as “logistic” growth,
which we will see is not predictable.
This system is an example of nonlinear
dynamics.
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Logistic Growth
Describes the behaviour of a population
that has limited resources
(food, water, space).
Growth of the population is limited by a
carrying capacity K.
The population increases, but becomes
saturated as it gets closer to the
carrying capacity forcing the rate of
growth to decrease.
Macquarie University 12
Effect of the Limit
We want to know how the population N
behaves when it gets “close” to the
carrying capacity K.
Will it level off and stabilise at N=K ?
N<K ?
Will it overshoot and settle back down?
Will it go into an oscillation?
Will it do something else?
Macquarie University 13
Logistic Growth Variables
How can we model this in Excel®?
Consider a population N and saturation level K
such that 0 ≤ N ≤ K.
Also introduce a variable x where:
N
x
K
Think of x as a “fraction of possible population”.
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e.g.
Suppose that for Australia, K = 100,000,000.
If the current population is Nn = 20,000,000
then:
Nn 1
xn 0.2
K 5
Of course, 0 ≤ xn ≤ 1 always
and the remaining capacity is 1 - xn.
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The Logistic
Difference Equation
Assume that the growth rate is not constant
but proportional to the
remaining capacity: r 1 xn
Growth rate term is now r (1-xn).
For small xn growth rate is ~r.
For large xn growth rate is ~0.
Population from generation n to generation
n+1 is given by: xn+1 = r (1-xn)xn .
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What is r ?
r remains as a parameter in the growth
rate term {r (1-xn) }, but r itself is a
variable.
Its lower bound is zero (if r=0,
population goes straight to zero; r<0 as
cannot have a “negative” population).
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The Growth Rate Term
If you multiply existing population xn by
1, you get back the same population
(stable).
If r (1-xn) < 1, the population will
decrease.
If r (1-xn) > 1, the population will
increase.
Is there an upper bound to r ?
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Let’s try r=1.5
Growth rate is 1.5(1-xn)
r = 1.5, N(0)=0.1, x(0)=0.1
4
3.5
3
Population
2.5
Geometric (N)
2
Logistic (x=N/K)
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9
Generations
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Population reaches equilibrium
When the growth rate is equal to 1.5
times the remaining population,
saturation pushes the population into
equilibrium at x=0.33.
Is equilibrium a normal condition for all
values of r ?
We have used an initial population
fraction of x0=0.1. What if we change
the initial population?
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Next try r=2.8
Growth rate is 2.8(1-xn)
r = 2.8
0.8
0.7
0.6
0.5 x(0)=0.1
x(n)
0.4 x(0)=0.2
0.3 x(0)=0.3
0.2
0.1
0
0 5 10 15 20 25
n
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An Attractor
It appears that no matter what initial
population x0 we start with, the
population reaches the same
equilibrium value (after transients die
out) for r=2.8.
When a population settles like this, for
any starting value, the eventual
behaviour is known as an attractor.
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r=3.14
Growth rate is 3.14(1-xn)
r = 3.14
0.9
0.8
0.7
0.6
x(0)=0.1
0.5
x(n)
x(0)=0.2
0.4
x(0)=0.3
0.3
0.2
0.1
0
0 5 10 15 20 25
n
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r=3.45
Growth rate is 3.45(1-xn)
r = 3.45
1
0.9
0.8
0.7
0.6
x(0)=0.2
x(n)
0.5
x(0)=0.3
0.4
0.3
0.2
0.1
0
0 5 10 15 20 25 30 35 40 45 50 55
n
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r=3.45
4-cycle
r = 3.45
1
0.9
0.8
0.7
0.6
x(n)
0.5 x=0.2
0.4
0.3
0.2
0.1
0
0 5 10 15 20 25 30 35 40 45 50 55
n
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r=3.8
Growth rate is 3.8(1-xn)
r = 3.8
1
0.9
0.8
0.7
0.6
x(0)=0.2
x(n)
0.5
x(0)=0.3
0.4
0.3
0.2
0.1
0
0 20 40 60 80 100
n
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Attractors
Attractors have different behaviours and
values depending on value of r.
r = 2.8 r = 3.14 r = 3.45 r = 3.8
equilibrium 2-cycle 4-cycle aperiodic
constant oscillates oscillates appears
random
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Mapping the Attractor
It can be shown mathematically that
r=4 is a limit for this model.
Can we create a map in Excel® that
displays the long-term behaviour of the
attractor for 0 ≤ r ≤ 4 ?
For each r, we can plot a sequence of
values of xn for large n (after transients
have died out).
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The Spreadsheet Formula
A B C D E F
1 n r 0 0.1 0.2 0.3
2 0 x(0) 0.1 0.1 0.1 0.1
3 1 =C$1*(1-C2)*C2
4 2
5 3
6 4
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Fill the Spreadsheet
A B C D E F
1 n r 0 0.1 0.2 0.3
2 0 x(0) 0.1 0.1 0.1 0.1
3 1 0 0.009 0.018 0.027
4 2 0 0.000892 0.003535 0.007881
5 3 0 8.91E-05 0.000705 0.002346
6 4 0 8.91E-06 0.000141 0.000702
7 5 0 8.91E-07 2.82E-05 0.00021
8 6 0 8.91E-08 5.63E-06 6.31E-05
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A B C D E F G H I J K
1 n r 0 0.5 1 1.5 2 2.5 3 3.5 4
2 0 x(0) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
3 1 0 0.045 0.09 0.135 0.18 0.225 0.27 0.315 0.36
4 2 0 0.021488 0.0819 0.175163 0.2952 0.435938 0.5913 0.755213 0.9216
5 3 0 0.010513 0.075192 0.216721 0.416114 0.61474 0.724993 0.647033 0.289014
6 4 0 0.005201 0.069538 0.254629 0.485926 0.592087 0.598135 0.799335 0.821939
7 5 0 0.002587 0.064703 0.28469 0.499604 0.6038 0.721109 0.561396 0.585421
8 6 0 0.00129 0.060516 0.305462 0.5 0.598064 0.603333 0.861807 0.970813
9 7 0 0.000644 0.056854 0.318233 0.5 0.600959 0.717967 0.416835 0.113339
10 8 0 0.000322 0.053622 0.325441 0.5 0.599518 0.607471 0.850793 0.401974
11 9 0 0.000161 0.050746 0.329294 0.5 0.60024 0.71535 0.444306 0.961563
12 10 0 8.04E-05 0.048171 0.331289 0.5 0.59988 0.610873 0.864144 0.147837
13 11 0 4.02E-05 0.045851 0.332305 0.5 0.60006 0.713121 0.410898 0.503924
14 12 0 2.01E-05 0.043749 0.332818 0.5 0.59997 0.613738 0.847213 0.999938
15 13 0 1.01E-05 0.041835 0.333075 0.5 0.600015 0.711191 0.453051 0.000246
16 14 0 5.03E-06 0.040084 0.333204 0.5 0.599992 0.616195 0.867285 0.000985
17 15 0 2.51E-06 0.038478 0.333269 0.5 0.600004 0.709496 0.402856 0.003936
18 16 0 1.26E-06 0.036997 0.333301 0.5 0.599998 0.618334 0.84197 0.015682
19 17 0 6.28E-07 0.035628 0.333317 0.5 0.600001 0.707991 0.465697 0.061745
20 18 0 3.14E-07 0.034359 0.333325 0.5 0.6 0.620219 0.870882 0.23173
21 19 0 1.57E-07 0.033178 0.333329 0.5 0.6 0.706642 0.393564 0.712124
22 20 0 7.85E-08 0.032078 0.333331 0.5 0.6 0.621897 0.83535 0.820014
23 21 0 3.93E-08 0.031049 0.333332 0.5 0.6 0.705423 0.481392 0.590364
24 22 0 1.96E-08 0.030085 0.333333 0.5 0.6 0.623404 0.873788 0.967337
25 23 0 9.82E-09 0.02918 0.333333 0.5 0.6 0.704315 0.385989 0.126384
26 24 0 4.91E-09 0.028328 0.333333 0.5 0.6 0.624767 0.829505 0.441645
27 25 0 2.45E-09 0.027526 0.333333 0.5 0.6 0.7033 0.494993 0.986379
28 26 0 1.23E-09 0.026768 0.333333 0.5 0.6 0.626008 0.874912 0.053742
29 27 0 6.14E-10 0.026051 0.333333 0.5 0.6 0.702366 0.383043 0.203415
30 28 0 3.07E-10 0.025373 0.333333 0.5 0.6 0.627144 0.827124 0.64815
31 29 0 1.53E-10 0.024729 0.333333 0.5 0.6 0.701503 0.500466 0.912207
32 30 0 7.67E-11 0.024117 0.333333 0.5 0.6 0.628189 0.874999 0.320342
33 31 0 3.84E-11 0.023536 0.333333 0.5 0.6 0.700703 0.382814 0.870893
34 32 0 1.92E-11 0.022982 0.333333 0.5 0.6 0.629155 0.826936 0.449754
Only plot 35 33 0 9.59E-12 0.022454 0.333333 0.5 0.6 0.699957 0.500894 0.989902
}
36 34 0 4.79E-12 0.02195 0.333333 0.5 0.6 0.630052 0.874997 0.039986
data after 37
38
39
35
36
37
0
0
0
2.4E-12
1.2E-12
5.99E-13
0.021468
0.021007
0.020566
0.333333
0.333333
0.333333
0.5
0.5
0.5
0.6 0.69926 0.38282 0.153548
0.6 0.630887 0.826941 0.519885
0.6 0.698606 0.500884 0.998418
transients 40
41
42
38
39
40
0
0
0
3E-13
1.5E-13
7.49E-14
0.020143
0.019737
0.019347
0.333333
0.333333
0.333333
0.5
0.5
0.5
0.6 0.631667 0.874997 0.006317
0.6 0.697991 0.38282 0.025107
0.6 0.632398 0.826941 0.097905
have 43
44
45
41
42
43
0
0
0
3.75E-14
1.87E-14
9.36E-15
0.018973
0.018613
0.018267
0.333333
0.333333
0.333333
0.5
0.5
0.5
0.6 0.697412 0.500884 0.353278
0.6 0.633086 0.874997 0.913891
0.6 0.696865 0.38282 0.314778
died out 46
47
44
45
0
0
4.68E-15
2.34E-15
0.017933
0.017611
0.333333
0.333333
0.5
0.5
0.6 0.633733 0.826941 0.862771
0.6 0.696347 0.500884 0.473588
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The Logistic Map
1.2
1
0.8
The Attractor
0.6
0.4
0.2
0
0 1 2 3 3.1 3.2 3.48 3.553 3.59 3.69 3.79 3.89 3.99
r
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The Logistic Map
1
0.9
0.8
0.7
0.6
The Attractor
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 3.1 3.2 3.48 3.553 3.59 3.69 3.79 3.89 3.99
r
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The Logistic Map
The Logistic Map looks the same for all
values of starting population fraction x0
(because the whole map is an attractor,
and we are looking at the long-term
behaviour).
But if we look at r=3.8, for example,
the values for x0=0.1 and x0=0.2 are
very different at later times.
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Sensitive Dependence on
Initial Conditions (SDOIC)
A small difference in the value of r or x0
can make a huge difference in the
outcome of the system at generation n
(“butterfly effect”).
No formula can tell us what x will be at
some specified generation n even if we
know the initial conditions.
The system is unpredictable!!
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Stephen Hawking:
“We already know the physical laws
that govern everything we experience
in everyday life … It is a tribute to how
far we have come in theoretical physics
that it now takes enormous machines
and a great deal of money to perform
an experiment whose results we cannot
predict.”
Macquarie University 36
CHAOS
The attractor branches into two, then four,
then eight and so on. The sequence follows a
geometric progression, but soon looks like a
mess.
Messy regions are cyclically interspersed with
clear “windows”.
Existence of period-3 windows implies chaos.
Macquarie University 37
Features of Chaos
Period 3 region.
Chaotic systems show self-similarity or
fractal behaviour.
SDOIC – points that start off close
together can be widely separated at a
later time (also referred to as “mixing”).
Macquarie University 38
Period-Doubling
Constant > period-two > period-4 >
period-8 > … > chaos > …
Bifurcations mark the transition from
order into chaos.
Bifurcations follow a pattern, occurring
closer and closer together, ad infinitum.
Look at their relative separations…
Macquarie University 39
Measuring Feigenbaum's Number
1
0.9
0.8
0.7 r1
r2
0.6
0.5 r3
0.4
0.3
0.2
0.1 this length this length
0
0 1 2 3 4
r
Macquarie University 40
Feigenbaum’s Constant
Feigenbaum’s constant is:
rn rn 1
lim 4.6692016609 10299097 ...
n 1 rn
n r
The Feigenbaum point is at
r=3.5699456…
Macquarie University 41
Universality in Chaos
Feigenbaum’s number is observed
in all chaotic systems.
Measured in physical systems:
Dripping taps.
Oscillation of liquid helium.
Fluctuation of gypsy moth populations.
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Another Chaotic System
The logistic map is a quadratic map in
one dimension – the one variable is
x(r).
Chaos can involve multi-dimensional
systems.
An example is the mapping that
generates the attractor of Hénon.
Macquarie University 43
Attractor of Hénon
Make two columns, one for x and one for y
values.
Can choose (0,0)
as starting point. xn 1 yn axn 1
2
Generate subsequent y
n 1 bxn
rows using formulae:
Changing parameters a 7 / 5
a and b will generate
different attractors. b 3 / 10
Macquarie University 44
Attractor with parameters
a=7/5, b=3/10
Attractor of Henon
0.5
0.4
0.3
0.2
0.1
0
y
-1.5 -1 -0.5 0 0.5 1 1.5
-0.1
-0.2
-0.3
-0.4
-0.5
x
Macquarie University 45
The 3-lane feature
Attractor of Henon
0.3
0.25
0.2
0.15
y
0.1
0.05
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
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Chaos is Everywhere
Perfect systems may be easily modelled
according to the “laws of physics”: with
the massless ropes, frictionless surfaces
and perfect vacuum of physics text-
book problems.
Real systems have friction, air-
resistance and physical variations that
make them unpredictable.
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Examples of Chaos
Laser instabilities.
Fluid turbulence.
Progression to heart attack.
Population biology.
Weather.
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Bifurcation = Branching
Branching is important for life:
Trees, but also blood vessels, nerves.
Clones are not identical:
Branches are not pre-determined;
DNA codes for branching capability;
Makes the code economical.
Non-living systems – lightning,
snowflakes.
Macquarie University 49
Landmark Publications
Lorentz, Edward N., Deterministic Nonperiodic Flow,
J. Atmos. Sci. 20 (1963) 130-141.
Li, Tien-Yien & Yorke, James A., Period 3 Implies
Chaos, American Mathematical Monthly 82 (1975)
343-344.
Hénon, Michel, A two-dimensional mapping with a
strange attractor, Comm. Math. Phys. 50 (1976) 69-
77.
May, Robert M., Simple mathematical models with
very complicated dynamics, Nature 261 (1976) 459-
467.
Feigenbaum, Mitchell J., Quantitative universality for
a class of nonlinear transformations, J. Stat. Phys. 19
(1978) 25-52.
Mandelbrot, Benoit B., Fractal aspects of the iteration
of z → lz(1-z) for complex l and z, Annals NY Acad.
Sciences 357 (1980) 249-257.
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Acknowledgements
This presentation was based on lecture
material for PHYS220 presented by
Prof. Barry Sanders, 2000-2003.
Additional References:
Peitgen, Jürgens & Saupe, Chaos and
Fractals: New Frontiers of Science, 1992.
Gleick, Chaos: Making a New Science,
1987.
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