idea for a function

[abskebabs]

Recently I had an idea of a function that I was interested in plotting after playing with my calculator. For some reason I feel it may have interesting properties and as it is based on the trigonometric funtion tan... it could be interesting extending it and start applying it to other geometric functions or series. Anyway, I will introduce my idea:

 

Say you have f(x)=tan(x). Now, once you have a result you take the tan of your answer. Therefore you have; f(f(x))=tan(f(x)). This could then be continued and the "order" n of the function would be the independent variable. The order of the function above for example would be n=2, and for the one above it; 1. What I have been wanting to do is to make a plot of this function with a range of "starting point" values of x on the conventional vertical axis of the graph, and on the horizontal axis to have it plotted against the order n. The results then could be displayed for each starting point and a rough "path" if any of the starting points yield patterns that appear to be continuous.

 

This could be visually amplified by using a graph with shortened divisions between the integer values of the order n on the horizontal axis. Basically, I am wondering if this function will display any intersting properties, and what special areas and starting points there are. I can already make an educated guess that there will be values towards which the results of the function may tend towards. Also there will be some that do not change, e.g. if you start at 0, u stay at zero, no matter what the order is.

 

I have tried, but do not know how to display this kind of plot using the software available to me(Mathcad), so I would be very grateful if someone could please try and do this this and tell me what they observe. I would be even more grateful if a plot could be posted on this thread.

 

Finally I would like to ask; is this idea original? If not could you direct me to more information about these types of functions. The operation, as I have already stated, could easily be extended, and the properties discovered could be interesting.

 

So what do you guys think having read that?

[Dave]

The function you've described could be quite interesting. Generally, these iterated functions can be investigated using cobweb diagrams. Take a look at the Logistic map, which is somewhat simpler than the function you're looking at and introduces some of the simpler ideas behind this sort of theory.

[abskebabs]

The function you've described could be quite interesting. Generally, these iterated functions can be investigated using cobweb diagrams. Take a look at the Logistic map, which is somewhat simpler than the function you're looking at and introduces some of the simpler ideas behind this sort of theory.

It's intriguing you say this, as I had suspected such a link as relatively recently I have taken a module in my course where we studied Chaos and Nonlinear systems. In fact I think it inspired this idea. The logistic map already has application in physics and applied mathematics. I guess I was also wondering if this specific function or something similiar could find itself in physical application. It would be intriguing if this were to be the case.

 

Also I was wondering, probably erroneously though, that If a the phase plot of this function was plotted, then if we looked at the path the function takes from a certain starting value as we increase the value of n, we could observe

perhaps "half" functions of tan or fractional functions of tanif we looked at the values produced between discrete integer values of n. I do feel this specific idea is slightly dubious though, and so I am not uite sure of its logical consistency.

 

Again I would like to reiterate, could someone please try producing a plot of this?

[Royston]

I have tried, but do not know how to display this kind of plot using the software available to me(Mathcad), so I would be very grateful if someone could please try and do this this and tell me what they observe. I would be even more grateful if a plot could be posted on this thread.

 

abskebabs, I'm currently using Mathcad, and have some handy worksheets for plotting functions as part of my course. If you could provide exactly what your plugging in to Mathcad, I might be able to see where you're going wrong. It would be interesting to see the long term behaviour, where the pattern tends to, and perhaps, unless you have already, evaluate a closed form.

[Dave]

Also I was wondering, probably erroneously though, that If a the phase plot of this function was plotted, then if we looked at the path the function takes from a certain starting value as we increase the value of n, we could observe

perhaps "half" functions of tan or fractional functions of tanif we looked at the values produced between discrete integer values of n. I do feel this specific idea is slightly dubious though, and so I am not uite sure of its logical consistency.

 

Again I would like to reiterate, could someone please try producing a plot of this?

 

I see where you're going with this, but I'm afraid that this is ill-defined. What you currently have is a function [math]f_{x_0} : \mathbb{N} \to \mathbb{R}[/math], defined by:

 

[math]f_{x_0}(n) = \underbrace{\tan \circ \cdots \circ \tan}_{n \text{ times}}(x_0)[/math]

 

And what you wish to do is extend the domain of this function from the natural numbers to all real numbers. Unfortunately, without some kind of special form for [math]f_{x_0}[/math], you can effectively pick any extension you like and it won't necessarily be the 'correct' extension. Please get back to me if you think I'm wrong.

 

For the second half of your question, you might wish to consider using something like Matlab or its open-source alternative, GNU Octave. They're pretty powerful and are probably capable of doing this sort of thing.

[abskebabs]

abskebabs, I'm currently using Mathcad, and have some handy worksheets for plotting functions as part of my course. If you could provide exactly what your plugging in to Mathcad, I might be able to see where you're going wrong. It would be interesting to see the long term behaviour, where the pattern tends to, and perhaps, unless you have already, evaluate a closed form.

 

I've not done much so far, just a simple x-y plot with the the "angle" on the horizontal axis plotted against several functions of tan, each with a different degree of iteration, as well as x itself. If you want you can have a look at it, I have attached a screenshot JPEG of what I did, as unfortunately I cannot upload Mathcad as it doesn't seem to be an allowed file type.

 

Nevertheless, I think you will find it quite interesting(I myself was pretty excited during the process ofdoing this:-) ). For example there are regions or "starting values" of x at which the iterated functins all join up as can be seen graphically, and do actually tend towards a certain value. There are also other regions which go through "cycles"(I don't think they're necessarily periodic in terms of the order of iterations but they do tend to go around the same range of values), whereby they may start off at a certain value of x,e.g. 0.3 , you keep using the tan function and it tends towards a certain value before making a jump to a different value(with the tan function this usually turns out to be -ve and close to -pi/2). A lot of this was found just messing around with my calculator, but I felt gladdened seeing it graphically. I will try and get up to scratch with cobweb diagrams, logistic maps and phase diagrams by reading up my chaos module, before I attempt to do anything more advanced. Also I will need to learn how to do these on Mathcad too!

 

If you know how to do this though I urge you to give it a try though. Pls tell me how you do. There is whole host of interesting behaviour I have not yet mentioned, but I leave it to interested observers to have a look at this and see for themselves.

 

EDIT: As you can see I've fiddled with the limits a bit to produce a raph with a more reasonable scale, as I'm not interested in where the functions tend towards infinity. Also you may notice that if you alter the division between values of x, it drastically changes the graphs you see, and the dependency grows as you increase the degree of bifurcation.

[abskebabs]

Also you may notice that if you alter the division between values of x, it drastically changes the graphs you see, and the dependency grows as you increase the degree of bifurcation.

OMG:D ! Have I just quoted myself observing that this function displays chaotic behaviour? It becomes increasingly susceptible to changes in initial conditions as the order of iteration grows! Pls corect me if I am mistaken, but if valid this would be rather interesting.. but I'm not sure of the validity...

[abskebabs]

It's been quite a while and only 2 ppl have replied to this thread at all. Is the subject matter not that interesting, or do ppl feel they do not have much to say on this as there has been quite a few views?

[Royston]

It's been quite a while and only 2 ppl have replied to this thread at all. Is the subject matter not that interesting, or do ppl feel they do not have much to say on this as there has been quite a few views?

 

Personally, I'm yet to study chaotic behaviour, so I don't really have anything to add concerning the functions validity, I think you need the advice of an expert.

[CPL.Luke]

its interestng that the first tangent line seems to yield an expoentil, however I may be missing parts of the line.