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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

 

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Both graphs superimposed 

 

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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

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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!

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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.