This is a high-precision computation of the accumulation point, lambda_inf, of the period 2^N cycles of the logistic map. Basic info about the accumulation point is presented at https://mathworld.wolfram.com/LogisticMap.html so I won't repeat it here.

This computation finds lambda_inf by computing the bifurcation point for increasing values of n. It finds the lambda_n by solving the nonlinear system,

x2 = lam * x1 * (1-x1)

x3 = lam * x2 * (1-x2)

...

x1 = lam * xn * (1-xn)

lam^n * (1-2 * x1) * (1-2 * x2) * .... * (1-2 * xn) = -1

where n=2, 4, 8, 16, etc is the number of fixed points. The first n equations are found by simply unrolling the logistic map iteration, and requiring that after n iterations I get the original fixed point back again. The last equation expresses the fact that at the bifurcation point, the slope of the overall iterated map f^(n)(x) = -1 for any fixed point x. Note that this system has n+1 equations and n+1 unknowns.

I solve this system using Newton's method. I first solve the n=2 cycle, then use the resulting x and lam as starting guesses for the n=4 Newton iteration. I repeat this procedure for n = 4, 8, 16, ... etc. The computation takes longer for increasing n. I can get up to around period 2^17 but after that Newton's method fails to converge and/or I run out of memory depending upon my precision and tolerance settings.

The computation uses MPFR for high-precision numerics, and Eigen to provide solvers and other linear algebra functions. The Jacobian matrix used in Newton's method is sparse to help save memory. A dynamically-sized sparse solver is used when solving for the Newton step.

Results of the computation are dumped into a file called "results.txt". After a run I move the results file into the "results" directory for post-processing. Specific post-processing steps involve checking how many digits match between different runs using different precision and tolerance settings. I also post-process using Aitken sequence acceleration to squeeze more digits out of the estimated value of lambda_inf. More information about that procedure is given in the results directory.

The code is written in easy C++. If you want to repeat these computations, you need to download and install the following libraries (and their dependencies):

• MPFR++: http://www.holoborodko.com/pavel/mpfr/
• Eigen: http://eigen.tuxfamily.org/index.php?title=Main_Page

Place both libraries into the main directory as sub-directories. Then do "make" and if all goes well you should have an executable called bifurcation_calc which you can play with. If you manage to compute lambda_n values for n > 2^17, please drop me a line and tell me how you did it.

Stuart Brorson 6.17.2020