Exploring Chaos Theory: Studying a Parameter to Uncover Its Behavior

[mlopes]

First of all, my sincere wishes of an Happy Christmas, full of joy, happiness and such feelings that make our life worthfull!

I'm studying and researching about Chaos Theory, and currently I'm examining the following iterative equation:

Xn+1 = c*(Xn-(Xn^3)/3)

I've already managed to give an approximation to which c values does the function get in chaos. But to have further details on this problem, i do have to study the equation for each different value of c parameter!

Which is the best way, analitically and graphically, to study and return the "exact" values of c that make the equation: converge, oscilate between to values or enter chaos?

Best Regards and Merry Christmas! :-)

 

[TillEulenspiegel]

I can't give you an absolute definition and limits of c , but the idea is that set c between upper and lower limits and the iteration will meander. I first saw this in Scientific American in the early 80's ( Mandlebrot sets and attractors ) and Lucas Arts developed a game " rescue on fractlius" for the commodore 64 that generated fractal geometry on the fly ... God That was such fun > I tried to develop weighted color maps of the expression of the algorithm's output but failed, it was fun tho for a kid = ).. There are some great math people here maybe they can help, there are also C and C++ routines that are available off the web where you can skip iterations and see the results quicker and are displayed in graphics. The upper and lower limits are easy to establish this way ( and will surly be faster on say a pent 4 @ 2 Ghz than a 6502 at 1 Mhz )

 

[mlopes]

Hi thanks for your kind reply! :-)

Actually, processing power isn't a problem at all. I suppose that it is impossible to calculate with full accuracy the correct values for the c parameter, but with 4/5 decimal places would be a great approach.
The problem is how to calculate the c value.. is there a program already made to do this? Which is the best way analitically to do it?

Best Regards and Wishes of a great 2004! :-)

 

[dhris]

Well, I would be inclined to just run the equation in Mathematica or Matlab (or even in Excel for that matter) and look at the behaviour that way. Then keep adjusting c until you find the boundary. You can easily find the attractor set analytically though by setting
$$x_{n+1}=x_n$$

dhris

 

[jamp46]

Originally posted by mlopes
First of all, my sincere wishes of an Happy Christmas, full of joy, happiness and such feelings that make our life worthfull!

I'm studying and researching about Chaos Theory, and currently I'm examining the following iterative equation:

Xn+1 = c*(Xn-(Xn^3)/3)

I've already managed to give an approximation to which c values does the function get in chaos. But to have further details on this problem, i do have to study the equation for each different value of c parameter!

Which is the best way, analitically and graphically, to study and return the "exact" values of c that make the equation: converge, oscilate between to values or enter chaos?

Best Regards and Merry Christmas! :-)

Please try thease little program
REM fiegenbaun tree
CLS
SCREEN 9
on error goto 100
COLOR , 7
WINDOW (-1, 0)-(3, 5)
FOR r = 0.005 TO 3 STEP .007
FOR i = 1 TO 200
x = .025
x = r * (x-x*x*x/3)
NEXT i

FOR j = 1 TO 200

x = r * (x-x*x*x/3)
PSET (x, r), 4

NEXT j
NEXT r
100 resume next

 

[jamp46]

Originally posted by mlopes
Hi thanks for your kind reply! :-)

Actually, processing power isn't a problem at all. I suppose that it is impossible to calculate with full accuracy the correct values for the c parameter, but with 4/5 decimal places would be a great approach.
The problem is how to calculate the c value.. is there a program already made to do this? Which is the best way analitically to do it?

Best Regards and Wishes of a great 2004! :-)

Dear mlopes, I have calculate your recurrent equation Lyapunov exponent and I have build Fiegenbaun tree. I thing, if is not some calculus error:
If c<2 is convergent and at c aprox= 2 are the first duplication ( f.e. if c= 1.500 converge to 1.0000, if 1.95 =>1,20894105, but if c= 2.05 => 1,082003562 and 1,352504452).
If c>= 2.35 chaos is developped, (exp. Lyap. >0) , about c= 2.75 is a little window of order and if c> 3 is divergent