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Thread Subject:
Bifurcation of Logistic Map

Subject: Bifurcation of Logistic Map

From: Sanaa

Date: 2 Apr, 2012 16:11:15

Message: 1 of 3

Hi,
I am still stuck with plotting the bifurcation diagram for the logistic map given by

x_n = ru*x_(n-1/2)*(1-x_(n-1/2))
where n = 1/2, 1, 3/2,....

any idea?

Subject: Bifurcation of Logistic Map

From: Roger Stafford

Date: 2 Apr, 2012 19:32:10

Message: 2 of 3

"Sanaa" wrote in message <jlcj33$7uh$1@newscl01ah.mathworks.com>...
> I am still stuck with plotting the bifurcation diagram for the logistic map given by
> x_n = ru*x_(n-1/2)*(1-x_(n-1/2))
> where n = 1/2, 1, 3/2,....
- - - - - - - - - - -
  In your two previous threads, #318401: "Index exceeds matrix Dimensions?", and #318570: "Problem with plot", you plotted the x-values in the logistic iteration x(n+1) = r*x(n)*(1-x(n)) starting all the way back to its initial value at n = 1 (although you called it n = 1/2 and 1/3.) However a bifurcation diagram only shows the values that x is cycling through periodically after a "settling down" period, and for this you should have gone far out in the iteration process before using the values in a plot. I think that explains your question in thread 318570 about having "so many lines in the beginning of my plot." Those are values you should not have been plotting for a valid bifurcation diagram.

  I would suggest a careful rereading of various articles about bifurcation diagrams. For example, there is one to be found at:

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html#BD

In that article there is the sentence, "For each value of r the system is first allowed to settle down and then the successive values of x are plotted for a few hundred iterations." Note carefully the "settle down" phrase!

  In your code you need to find out how far out to go in order for the sequence to "settle down", and then you should plot just those values the sequence is cycling through from there on, and to do all this for every value of r in your plot. Note that in the chaotic region of values for r>3.569 the values of x you get will typically be spread continuously over sizable intervals of x values.

Roger Stafford

Subject: Bifurcation of Logistic Map

From: Sanaa

Date: 2 Apr, 2012 23:50:13

Message: 3 of 3

Thanks a lot sir, I have been able to get the desired plot. I had to through away some values of x to obtain the bifurcation diagram. I appreciate your help.

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