You are currently browsing the tag archive for the ‘henon’ tag.

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.

Website Stats

  • 8,008,039 views

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 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.

Modelling Guide


IB Exploration Modelling Guide 

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

Available to download here.

Statistics Guide

IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis and Applications students!

Available to download here.

Recent Posts

Follow IB Maths Resources from British International School Phuket on WordPress.com