How do you find fixed points?

Regards,

]]>John,

This question is meaningless without reference to an iterator function. If is a differentiable function then is an attracting fixed point if etc

Regards,

J.P.

f(x)= x +sin x

How do you show that f(x) has an infinite number of fixed points? ]]>

Hi J.P, Could you show me how to show points like e^3(pi)i/4 are attracting or repelling or nuetral? I am a little bit stuck with this. Thanks

]]>Aoife,

Call the mortality rate by . Note this is a different to the one in the notes — which is the maximum population: more on this below. Therefore the equation governing the population growth is given by

.

You can solve then for the fixed points.

Note that is the proportion of the maximum population: the population that, if reached, results in extinction. Therefore if , i.e. at the maximum, then the next population is

.

The term models this.

Regards,

J.P.

Linda,

Yes you are correct to say that has lines.

A fixed point of a function is found at the intersection of and . Therefore draw a schematic (rough) graph of with its lines and . How many times does intersect the graph of ?

A second, more algebraic, solution is as follows. A fixed point of is a point that is sent to itself

.

Hence fixed points of are solutions of

;

in other words period- points. We know that the period- points are points of the form (why?)

(in binary)

and there are of these (why?), hence has fixed points.

Regards,

J.P.

First of all we are only taking for the Tent Mapping and call this . I suppose the questions that I could ask about the Tent Mapping include how many period- points does have, show that the periodic points are dense and show that any fraction is eventually periodic under .

Does your typeset notes stop at page 8 of the complex numbers section? Then you are not missing anything: I threw out a lot of the complex number stuff.

To your final question no and no — although we did all of the Autumn 2012 questions in the lectures and the CA test material is very straigtforward. However a word of caution: learning off answers is no good to ye now: ye need to understand the material.

Regards,

J.P.

For Question two I have all the proofs for Logistic Mappings and Doubling but have no proof/theorem for Tent Mapping. Is the theory for this section about the conditions of it which are 0<u<4 etc.

As regards the Complex Number section I have all the class written notes but seem to be missing typed notes in my booklet. Do you by any chance have an online PDF or e-copy for this section?

Finally are there solutions for the 2012 Autumn Question 3 and 4 and the Continuous Assessment Exam available?

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