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Chaos and strange Attractors: Henon’s map

Henon’s map was created in the 1970s to explore chaotic systems.  The general form is created by the iterative formula:

The classic case is when a = 1.4 and b = 0.3 i.e:

To see how points are generated, let’s choose a point near the origin.  If we take (0,0) the next x coordinate is given by:

We would then continue this process over several thousands iterations.  If we do this then we get the very strange graph at the top of the page – the points are attracted to a flow like structure, which they then circulate round.  The graph above was generated when we took our starting coordinate as (0.1,0.1), let’s take a different starting point.  This time let’s have (1.1, 1.1):

  We can see that exactly the same structure appears.  All coordinates close to the origin will get attracted to this strange attractor – except for a couple of fixed points near the origin which remain where they are.  Let’s see why.  First we can rewrite the iterative formula just in terms of x:

Next we use the fact that when we have a fixed point the x coordinate (and y coordinate) will not change.  Therefore we can define the following:

This allows us to then make the following equation:

Which we can then solve using the quadratic formula:

Which also gives y:

So therefore at these 2 fixed points the coordinates do not get drawn to the strange attractor.

Above we can see the not especially interesting graph of the repeated iterations when starting at this point!

But we can also see the chaotic behavior of this system by choosing a point very close to this fixed point.  Let’s choose (0.631354477, 0.631354477) which is correct to 9 decimal places as an approximation for the fixed point.  

We can see our familiar graph is back.  This is an excellent example of chaotic  behavior – a very small change in the initial conditions has created a completely different system.

This idea was suggested by the excellent Doing Maths With Python – which is well worth a read if you are interested in computer programing to solve mathematical problems.

Screen Shot 2018-05-16 at 10.36.10 AM

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).

Screen Shot 2018-05-16 at 10.12.48 AM

Starting value of x = 0.2

Screen Shot 2018-05-16 at 10.10.41 AM

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).

Screen Shot 2018-05-16 at 10.21.55 AM

Starting value of x = 0.2

Screen Shot 2018-05-16 at 10.25.52 AM

Starting value of x = 0.2001

Screen Shot 2018-05-16 at 10.23.03 AM

Both graphs superimposed 

Screen Shot 2018-05-16 at 10.23.14 AM

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:

Screen Shot 2018-05-16 at 10.29.49 AM

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.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

Screen Shot 2018-03-19 at 4.42.05 PM.png

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank 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 2021-05-19 at 10.05.18 AM

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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

Screen Shot 2018-05-16 at 7.45.04 AM

Both graphs superimposed 

Screen Shot 2018-05-16 at 7.45.31 AM

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

Screen Shot 2018-05-16 at 10.51.49 AM

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

Screen Shot 2018-05-16 at 10.57.01 AM

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:

Screen Shot 2018-05-16 at 12.02.53 PM

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.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

Screen Shot 2018-03-19 at 4.42.05 PM.png

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank 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 2021-05-19 at 10.05.18 AM

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

Screen Shot 2021-05-19 at 6.32.13 PM

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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?

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

Screen Shot 2018-03-19 at 4.42.05 PM.png

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank 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 2021-05-19 at 10.05.18 AM

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

Screen Shot 2021-05-19 at 6.32.13 PM

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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