my conjecture...will someone help me prove

[bishnu]

okay i have recently become intrested in iterative functions and im trying to solve the family of equations [math]x_{y+1}->x_y^n+c[/math]...k so using my calculator i graphed the equations i found some compelling results(btw trying to solve algebratically is tedious i belive it would be impossible tofind its limit)the y axes i set as the value the function converges too and x axes a value of c and i set x0 to zero i found that the functions stick to the function y=x and then suddenlystart to diverge from it,but as n increases the graph stay more and moreon y=x so my conjecture is that as n->infinity [math]x_{y+1}->x_y^n+c[/math] becomes the graph y=x through an intervel of about [0,.9]...my second connjecture is that the last convergent value of x_final of any n lies on a solution line of y equals about .7x+.27....

[matt grime]

Your conjecture is doesn't depend on n or C, are you sure that's what you want?

 

If n>1

If c=0 the 'function' obtained by iterating repeatedly connverges pointwise to zero for x between -1 and 1, is 1 for x=1 and =/-1 for x=-1 and diverges for all other real x.

 

if c is greater than 1 it divereges for all positive x, and if n is even diverges for all x.

 

If n=1 it diverges for all x,

 

htink abuot the cases a bit more.

[bishnu]

No that is not what I am trying to say. I'm sorry i must have explained it poorly. [math]x_{n+1}->x^m_n+c[/math](this is the family of functions i am studing)This is a form for an iterated function. Now what i am intrested in is the value this function converges too for each value of c and m...the x doesnt matter at all because it startes at zero but is only there to show how to iterated the function. Now what i did was i set up a graph with the x axis set as a value of c and the y the convergent value as the function is iterated toward infinity. Now each graph would represent 1 value of m. Now what i noticed about these functions was that the values stayed very close to the graph [math]y=x[/math]and then would suddenly start to diverge from the line. As the value om increased the graph stayed more and more on the line, so that leads to my first conjecture that as m goes to infinity the iterated function becomes that graph till it suddenly diverges around .9(ill explain how i got that value later) anotherthing i noticed was the maxium values of all the graphs seemed to lie close to the line [math]y=.7x+.2716..[/math] which if you find where they intersect you get the maxium value. My second conjecture is that all maxium values lie on that line.

I am only talking about graphes 1<m.

[Woxor]

It really seems counterintuitive that the values would initially approximate a line. I can definitely believe that they start out "under" it and then cross through it when they rapidly diverge, but perhaps they only seem to lie so close to the line because it's hard to obtain good resolution at that scale.

 

Also, I don't know about your "maximum values" here; there shouldn't be any [math]n[/math] at which [math]x_n[/math] is undefined or infinite -- in other words, you shouldn't have any vertical assymptotes, if that's what you're talking about. You'll probably just have really rapid divergence (something in [math]\Theta(e^{mx})[/math], if I had to take a guess). I don't think your conjecture is true, as I understand it, though I very well may not.

[bishnu]

program something to graph the above functions it actually does stay on the line

[Woxor]

Are you saying it diverges at n=0.9? If so, how are you interpolating the fractional values of n? If you're saying it diverges at x=0.9, then to be on the y=x (or in this case x=n) line, n would have to be 0.9, also, so either way, you need interpolation. How are you doing this?

 

EDIT: Also, I still don't know what you mean by the maximum value, or whether you're saying it diverges assymptotically.

[123rock]

The only way that this equation works for is if n and C are both constants of 1, and X sub Y increases by one for y+1, when y starts out at 0

[bishnu]

What does interpolating mean? But anyway the fractional values diverge similarly to the the integer values of n. My graph does not diverge asytomically. Since each y on the graph repersent s the corresponding convergent value of any a c then at a certain part it just stops(kind of like how arcsin just stops). I think you are a bit confused about my graphing...each x does not correpsond to a n but to the constat c.

Can you post pictures on here or do you have to link to them?

[bishnu]

123rock i think you have misunderstood what i am trying to say

[matt grime]

You appear not to have explained yourself very clearly.

 

Are you saying that yuo are defining f(z) for z in R to be the limit of the sequence

 

x_{n+1} = (x_n)^y+c with x_0=0

 

or what? what is the index over which you're iterating? x,n,y,m you've used all of them and swapped the use of n in the posts too. So try explaining it more clearly.

[MandrakeRoot]

Yeah i didnt quite catch what you are doing. Are you trying to plot the evolution of a sequence ?

What is your conjecture exactly ? If you formulate it very precisely you might be able to proof it !

Try formulating it like : "Let x be .....". If ..... ,then ..... or something like that.

 

Mandrake