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Test

The test will now take place on Monday February 18. Everything up to but not including section 3.4 in the typeset notes is examinable. We have everything for the test covered. Please find a sample test and the test I gave last year in your typeset notes.

Week 4

In week 4 we introduced the idea of a cob-web diagram, which illustrates the dynamics of points near fixed points.

We showed that if we take an iterator function of the form f(x)=mx+c, a line with slope m>0, then the fixed point of this iterator function is either attracting or repelling. 

A fixed point x_f\in S is an attracting fixed point if there exists an interval I containing x_f such that all orbits that begin in I converge to x_f.

A fixed point x_f\in S is a repelling fixed point if there exists an interval I containing x_f such that all orbits that begin in I eventually leave I.

As differentiable functions are approximated well by lines, we argued that if the slope of an iterator function is less than one (although positive) near a fixed point that this fixed point might be attractive.

We then outlined a soft version of the Contraction Mapping Principle:

Suppose that f:S\rightarrow S is a contraction mapping defined on a suitably nice set S. Then f has a unique attracting fixed point.

A contraction mapping is a function that sends points in S that are close to each other to points that are even closer to each other:

|f(x)-f(y)|<|x-y| for all x,\,y\in S.

 Nice here means compact which is beyond the scope of the course: roughly, for the purposes of MS3011, it means that all sequences in S that converge do so to a point in S and that the set is somehow not of infinite extent.

We didn’t give a proof but instead argued why the Theorem should be true. We showed that if you kept applying f(x) to the set S then the space shrinks to a point which must necessarily be fixed.

Inspired by this we proved the following theorem:

Suppose that f:\mathbb{R}\rightarrow \mathbb{R} is a (continuously) differentiable function with a fixed point x_f\in S. Then 

  • If |f'(x_f)|<1 then x_f is an attracting fixed point of f
  • If |f'(x_f)|>1 then x_f is a repelling fixed point of f
  • If |f'(x_f)|=1 then no condition can be drawn and we call x_f neutral or indifferent.

We have now covered enough to answer questions one and three on the final exam and everything on the test.

What next?.

Now we begin our special case study of three families of dynamical systems the first of which is the Logistic Family.

Exercises

Exercises for Thursday 7 February are to look at the following:

Sample Test & Actual Test in written notes

Autumn 2012: Question 3 (b)

Summer 2011: Question 1 (don’t worry about classifying the period-2 points). Question 2 would be a good preparation for the next few lectures also.

Tutorial Venue

I have applied to get this changed to WGB G03. Expect an email this week…

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