Summary of lecture 7
• Error detection schemes arise in a range of different places, such as
• Travelers Checks
• airline tickets
• bank account numbers
• universal product codes (barcodes)
• ISBNs
• Zip codes.
• In these cases, check digits are used to ensure that codes are correct.
• If an error arises we are able to correct the error manually.
MAT199: Math Alive
Birth, Growth, Death and Chaos
Ian Griffiths
Mathematical Institute, University of Oxford,
Department of Mathematics, Princeton University
Birth, Growth, Death and Chaos
• Dynamical systems are mathematical models for phenomena that change over
time.
Birth, Growth, Death and Chaos
• Dynamical systems are mathematical models for phenomena that change over
time.
• Examples include population dynamics:
• How do epidemics arise?
• How does human intervention change natural cycles?
e.g., how can we safely harvest fish without causing extinction?
what are the best measures to take to combat illness outbreaks?
Birth, Growth, Death and Chaos
• Dynamical systems are mathematical models for phenomena that change over
time.
• Examples include population dynamics:
• How do epidemics arise?
• How does human intervention change natural cycles?
e.g., how can we safely harvest fish without causing extinction?
what are the best measures to take to combat illness outbreaks?
• In this topic we will study mathematical modelling – rather than making the
world fit into our mathematical construction we use mathematics to describe the
world around us.
Investment strategies
1. Bury your money
Savings ($)
200
150
100
50
2
4
6
Number of years
• If P0 is originally invested then:
Investment after year n,
Pn =P0
8
10
Investment strategies
2. Simple interest
• Each year the bank pays you a fixed fraction, r, of your original investment.
Savings ($)
140
120
100
80
60
40
20
2
4
6
8
Number of years
• If P0 is originally invested then:
Investment after year n,
Pn =(1+nr) P0
10
Investment strategies
3. Compound interest
• Each year the bank pays you a fraction, r, of the investment you have that year.
Savings ($)
Number of years
• If P0 is originally invested then:
Investment after year n,
Pn =(1+r)n P0
Investment strategies
3. Compound interest
• Each year the bank pays you a fraction, r, of the investment you have that year.
Savings ($)
Number of years
• If P0 is originally invested then:
Investment after year n,
Pn =(1+r)n P0
Investment strategies
4. Compound interest and saving each year
• Each year the bank pays you a fraction, r, of the investment you have that year
and you save S dollars each year too.
• If P0 is originally invested then:
Investment after year n,
Pn =(1+r)n P0 + ( 1+(1+r)+(1+r)2 +(1+r) 3+…) S
Investment strategies
4. Compound interest and saving each year
• Each year the bank pays you a fraction, r, of the investment you have that year
and you save S dollars each year too.
• If P0 is originally invested then:
Investment after year n,
Pn =(1+r)n P0 + ( 1+(1+r)+(1+r)2 +(1+r) 3+…) S
How can we write this more
compactly?
Summary of lecture 8
• Mathematical modelling is used to describe and understand the world around us.
• Dynamical systems are mathematical models for phenomena that change over
time.
• We can write down rules that tell us what state we are in at any given time,
e.g., the amount of savings we have in a bank account at any given time.
• It is easy to work out the state we will be in at the next step (e.g., the money we
have in our account next year) given the current state.
• It is helpful if we can obtain expressions for the state we will be in at any time in
terms of the original state.
Carl Friedrich Gauss
1777 – 1855
Geometric progressions
sum
2.0
1.5
1.0
0.5
2
4
6
8
10
12
14
n
Geometric progressions
sum
2.0
1.5
1.0
0.5
2
4
6
8
10
12
14
n
Geometric progressions
• If -1<a<1 this result tells us that
which means that we can add an infinite number of terms and get a finite answer.
Summary of lecture 9
• We can use our ideas for investment strategies to write down mathematical
models to describe and understand population growth.
• A model Pn+1 = (1+r) Pn tells us the population at time n+1 given the population
at time n.
Summary of lecture 9
• We can use our ideas for investment strategies to write down mathematical
models to describe and understand population growth.
• A model Pn+1 = (1+r) Pn tells us the population at time n+1 given the population
at time n.
• If r>0 we get exponential population growth (e.g., bacteria).
• If r<0 the population will die out.
Summary of lecture 9
• We can use our ideas for investment strategies to write down mathematical
models to describe and understand population growth.
• A model Pn+1 = (1+r) Pn tells us the population at time n+1 given the population
at time n.
• If r>0 we get exponential population growth (e.g., bacteria).
• If r<0 the population will die out.
• This model is not so realistic as it does not account for additional effects such as
food resources. A better model has a growth rate that depends on the population.
• This leads to the logistic map:
Models for population growth
1. No limits on growth
• Malthus: Pn+1 = (1+r) x Pn
Pn = (1+r)n x P0
Thomas Malthus FRS
1766 – 1834
Models for population growth
1. No limits on growth
• Malthus: Pn+1 = (1+r) x Pn
Pn = (1+r)n x P0
Models for population growth
1. No limits on growth
• Malthus: Pn+1 = (1+r) x Pn
Pn = (1+r)n x P0
P
60
50
40
30
20
10
1
2
3
4
n
P0=1, r=2
5
6
Models for population growth
1. No limits on growth
• Malthus: Pn+1 = (1+r) x Pn
Pn = (1+r)n x P0
P
P
1.0
60
0.8
50
40
0.6
30
0.4
20
0.2
10
1
2
3
4
n
P0=1, r=2
5
6
1
2
3
4
n
P0=1, r=-0.5
5
6
Models for population growth
1. No limits on growth
• Malthus: Pn+1 = (1+r) x Pn
Pn = (1+r)n x P0
P
P
1.0
60
0.8
50
40
0.6
30
0.4
20
0.2
10
1
2
3
4
5
6
n
P0=1, r=2
• In general:
if r>0 population grows unboundedly
if r<0 population dies out.
1
2
3
4
n
P0=1, r=-0.5
5
6
Easter Island
Population models: 2. Limits on growth
• Food resources finite so growth rate depends on number of species:
Rate,
Population models: 2. Limits on growth
• Food resources finite so growth rate depends on number of species:
Rate,
• In this case we have
• This is called the logistic map.
Population models: 2. Limits on growth
• e.g. 1: r0 = 2
Population models: 2. Limits on growth
• e.g. 1: r0 = 2
Population models: 2. Limits on growth
• e.g. 1: r0 = 2
Population models: 2. Limits on growth
• e.g. 1: r0 = 2
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Population models: 2. Limits on growth
• e.g. 1: r0 = 2
Pn+1
P3
>
^
P
>
2
^
P1
>
^
P
0
P1
P2
P
3
P
n
Population models: 2. Limits on growth
• e.g. 1: r0 = 2
Pn+1
This is a stable
equilibrium point
P3
>
^
P
>
2
^
P1
>
^
This is an unstable
equilibrium point
P
0
P1
P2
P
3
P
n
Population models: 2. Limits on growth
• e.g. 2: r0 = -1
2.0
1.5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Population models: 2. Limits on growth
• e.g. 2: r0 = -1
Pn+1
decay
r<0
...
P2
P1
(1+r) P
P0
Pn
Population models: 2. Limits on growth
• e.g. 2: r0 = -1
Pn+1
decay
r<0
(1+r) P
This is a stable
equilibrium point
...
P2
P1
P0
Pn
Population models: 2. Limits on growth
• e.g. 3
>
v
^
This is a limit cycle
<
q1
q2
Summary of lecture 10
• We had a lot of snow.
• We found that for the logistic map the only
way to calculate the population evolution
was to substitute in manually which is
boring.
• But we found a nice graphical way of
visualizing this.
1.45
1.58
• We saw the Fibonacci number sequence:
1, 1, 2, 3, 5, 8, 13, 21,…
• As we take the ratio of any number with the previous one we get closer and
closer to the number 1.61803… =
• This is called the Golden Ratio.
• We saw how beautiful people had ratios of their features that were close to
the Golden Ratio.
Models for population growth
The tent map
pn +1
1
0.75
x
n+1
0.5
0.25
0
0
0.25
0.5
x
pn =
0.75
n
2pn
if 0 ≤ pn ≤ ½
2(1-pn )
if ½ ≤ pn ≤ 1
1
pn
Question
Is a limit cycle in the tent map stable, unstable or neutrally
stable?
(Hint – think about the gradient of the graph)
pn +1
1
0.75
x
n+1
0.5
0.25
0
0
0.25
0.5
x
2pn
pn =
n
2(1-pn )
0.75
if 0 ≤ pn ≤ ½
if ½ ≤ pn ≤ 1
1
pn
The tent map
• Since the steepness of the tent map exceeds 1, any limit cycle or equilibrium
point will be unstable.
pn +1
1
0.75
x
n+1
0.5
0.25
0
0
0.25
0.5
x
pn =
0.75
n
2pn
if 0 ≤ pn ≤ ½
2(1-pn )
if ½ ≤ pn ≤ 1
1
pn
Summary of lecture 11
Stability of equilibrium points
• If the steepness (positive or negative) of the population graph is less than 1
at an equilibrium point then the equilibrium point is stable.
• If the steepness (positive or negative) of the population graph exceeds 1 at
an equilibrium point then the equilibrium point is unstable.
• We can have limit cycles when the steepness (positive or negative) of the
population graph equals 1.
• We can change variables to express a dynamical system in a different way.
e.g., instead of population of animals, the cost of looking after the animals.
Ingredients of Chaos
Chaos is defined by three features:
1) Sensitive dependence on initial
conditions.
2) Dense number of possible places
to visit.
3) Dense number of actual places
available to visit.