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Modelling more Chaos

This post was inspired by Rachel Thomas’ Nrich article on the same topic.  I’ll carry on the investigation suggested in the article.  We’re going to explore chaotic behavior – where small changes to initial conditions lead to widely different outcomes.  Chaotic behavior is what makes modelling (say) weather patterns so complex.

f(x) = sin(x)

This time let’s do the same with f(x) = sin(x).

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Starting value of x = 0.2

 

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Starting value of x = 0.2001

 

Screen Shot 2018-05-16 at 10.10.48 AM

 

Both graphs superimposed 

 

Screen Shot 2018-05-16 at 10.10.48 AM

This time the graphs do not show any chaotic behavior over the first 40 iterations – a small difference in initial condition has made a negligible difference to the output.  Even after 200 iterations we get the 2 values x = 0.104488151 and x = 0.104502319.

f(x) = tan(x)

Now this time with f(x) = tan(x).

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Starting value of x = 0.2

 

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Starting value of x = 0.2001

 

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Both graphs superimposed 

 

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This time both graphs remained largely the same up until around the 38th data point – with large divergence after that.  Let’s see what would happen over the next 50 iterations:

 

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Therefore we can see that tan(x) is much more susceptible to small initial state changes than sin(x).  This makes sense by considering the graphs of tan(x) and sin(x).  Sin(x) remains bounded between -1 and 1, whereas tan(x) is unbounded with asymptotic behaviour as we approach pi/2.

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Screen Shot 2018-05-16 at 10.36.10 AM

Modelling Chaos

This post was inspired by Rachel Thomas’ Nrich article on the same topic.  I’ll carry on the investigation suggested in the article.  We’re going to explore chaotic behavior – where small changes to initial conditions lead to widely different outcomes.  Chaotic behavior is what makes modelling (say) weather patterns so complex.

Let’s start as in the article with the function:

f(x) = 4x(1-x)

We can then start an iterative process where we choose an initial value, calculate f(x) and then use this answer to calculate a new f(x) etc. For example when I choose x = 0.2, f(0.2) = 0.64. I then use this value to find a new value f(0.64) = 0.9216. I used a spreadsheet to plot 40 iterations for the starting values of x = 0.2 and x = 0.2001. This generated the following spreadsheet (cut to show the first 10 terms):

Screen Shot 2018-05-16 at 7.44.29 AM

I then imported this table into Desmos to map how the change in the starting value from 0.2 to 0.2001 affected the resultant graph.

Starting value of x = 0.2

Screen Shot 2018-05-16 at 7.45.17 AM

Starting value of x = 0.2001

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Both graphs superimposed 

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We can see that for the first 10 terms the graphs are virtually the same – but then we get a wild divergence, before the graphs seem to synchronize more closely again.  One thing we notice is that the data is bounded between 0 and 1.  Can we prove why this is?

If we start with a value of x such that:

0<x<1.

then when we plot f(x) = 4x – 4x2 we can see that the graph has a maximum at x = 1/2:
Screen Shot 2018-05-16 at 9.34.26 AM.

Therefore any starting value of x between 0 and 1 will also return a new value bounded between 0 and 1.  Starting values of x > 1 and x < -1 will tend to negative infinity because x2 grows much more rapidly than x.

f(x) = ax(1-x)

Let’s now explore what happens as we change the value of a whilst keeping our initial starting values of x = 0.2 and x = 0.2001

a = 0.8

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both graphs are superimposed but are identical at the scale we are using.  We can see that both values are attracted to 0 (we can say that 0 is an attractor for our system).

a = 1.2

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Again both graphs are superimposed but are identical at the scale we are using.  We can see that both values are attracted to 1/6 (we can say that 1/6 is an attractor for our system).

In general, for f(x) = ax(1-x) with -1≤x≤1, the attractors are given by x = 0 and x = 1 – 1/a, but it depends on the starting conditions as to whether we will end up being attracted to this point.

f(x) = 0.8x(1-x)

So, let’s look at f(x) = 0.8x(1-x) for different starting values 1≤x≤1.  Our attractors are given by x = 0 and x = 1 – 1/0.8 = -0.25.

When our initial value is x = 0 we remain at the point x = 0.

When our initial value is x = -0.25 we remain at the point x = -0.25.

When our initial value is x < -0.25 we tend to negative infinity.

When our initial value is  -0.25 < x ≤ 1 we tend towards x = 0.

Starting value of x = -0.249999:

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Therefore we can say that x = 0 is a stable attractor, initial values close to x = 0 will still tend to 0.

However x = -0.25 is a fixed point rather than a stable attractoras

x = -0.250001 will tend to infinity very rapidly,

x = -0.25 stays at x = -0.25.

x = -0.249999 will tend towards 0.

Therefore there is a stable equilibria at x = 0 and an unstable equilibria at x = -0.25.

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Langton's Ant

This is another fascinating branch of mathematics – which uses computing to illustrate complexity (and order) in nature.  Langton’s Ant shows how very simple initial rules (ie a deterministic system) can have very unexpected consequences.  Langton’s Ant follows two simple rules:
1) At a white square, turn 90° right, flip the color of the square, move forward one unit
2) At a black square, turn 90° left, flip the color of the square, move forward one unit.

The ant exists on an infinite grid – and is able to travel N,S,E or W. You might expect the pattern generated to either appear completely random, or to replicate a fixed pattern. What actually happens is you have a chaotic pattern for around 10,000 iterations – and then all of a sudden a diagonal “highway” emerges – and then continues forever. In other words there is emergent behavior – order from chaos. What is even more remarkable is that you can populate the initial starting grid with any number of black squares – and you will still end up with the same emergent pattern of an infinitely repeating diagonal highway.

See a JAVA app demonstration (this uses a flat screen where exiting the end of one side allows you to return elsewhere – so this will ultimately lead to disruption of the highway pattern)

game of life

Such cellular automatons are a way of using computational power to try and replicate the natural world – The Game of Life is another well known automaton which starts of with very simple rules – designed to replicate (crudely) bacterial population growth. Small changes to the initial starting conditions result in wildly different outcomes – and once again you see patterns emerging from apparent random behavior. Such automatons can themselves be used as “computers” to calculate the solution to problems. One day could we design a computer program that replicates life itself? Could that then be said to be alive?

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

IB Exploration Modelling and Statistics Guide


IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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