How can I draw a bifurcation diagram of the logistic dynamical coupled logistic map lattice system in MATLAB?

The logistic dynamicalcoupled logistic lattice system is

xn+1(i) = (1 − L(e))f (xn(i)) + (L(e)/2)(f (xn(i − 1)) + f (xn(i + 1)))

Here, L(e) = 3.99*e*(1 − e), 0 ≤ e ≤ 1 and f(x) is the logistic map functon

Answer 1

[William Rose]

@Harpreet,

What have you done so far?

Your function L(e) is the logistic map function, with r=3.99. You have not specified the r-value for the logistic map function f(x)=r*x*(1-x), where 0<r<4. Therefore your system has two adjustabole parameters: r and e. Do you want to make a bifurcation diagram for r (at a fixed e) or for e (at a fixed r)?

You make the logistic map by simulating the system until it reaches a limit cycle, or until you give up, for many values of the parameter (r or e). Then you plot the parameter (r or e) on the x-axis, and the corresponding elements of the limit cycle as the corresponding y-values.

[Harpreet]

Thank you so much sir for your reply. Here, e is fixed and i want to plot the parameter r on x-axis and corresponding sequence on y-axis. I want to make graph like below for different lattices ( here μ is used in place of r):

[William Rose]

@Harpreet, you are welcome.

Is there an error in your initial posting? You wrote:

"xn+1(i) = (1 − L(e))f (xn(i)) + (L(e)/2)(f (xn(i − 1)) + f (xn(i + 1))) "

and I wonder if the left hand side should be

"xn(i+1) = ..."

If the original posting is correct, then I do not understand the meaning of the left hand side.

[William Rose]

Here is a script that makes a bifurcation diagram for the standard logistic map. You will have to modify it for your lattice system.

For each value of r, the script does Nsim iterations of the logistic map. It saves the final Nm values, at that particular value of r, for plotting later. Suppose Nsim=2000 and Nm=200. Then, when the r-value is chosen so that the system is in a limit cycle with period 4, the bifurcation diagram will appear to have four points at that partiucular r-value, but in fact each of the four points will be plotted 50 times, since the last 200 iterations will visit each of the four points 50 times. Likewise, if the r-value is chosen so that the system is chaotic, then the bifurcation diagram will show 200 distinct points at that r-value, which correspond to the final 200 iterations.

The blue vertical lines in the plot are bifurcation points, or guesses for bifurcation points.

The advantage of the plotting approach above is that there is no need to determine if you are in a limit cycle, nor is there a need to determine the period of the limit cycle.

[Harpreet]

Thank you sir.

[William Rose]

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