by Bruce Camber (an emerging document, started in December 2019)
Open up period doubling bifurcation.
Get to know the Feigenbaum constant:
δ = 4.669 201 609 102 990 671 853 203 821 578
Period doubling Bifurcation. In 1885 Henri Poincaré used the word, bifurcation, for the very first time. In 1975 mathematician, Mitchell Feigenbaum, discovered a limiting ratio for each bifurcation interval. It is a constant. The plain vanilla version of period doubling was guided by such constants, yet, from my initial studies, nobody could discern why. Nobody was looking at the 202 notations, so certainly nobody was looking below that 64th (or perhaps the 50th) notation.
There is nothing simple about Henri Poincaré and his work. Yet, when seen from the Planck scale and through the most simple-but-dynamic sphere, an aether that Poincaré may have well envisioned appears to manifest (p.7, The Foundations of Science: Science and Hypothesis…, Science Press, 1913 by H. Poincaré, authorized translation by George Bruce Halsted).
Jules Henri Poincaré brings so much to this exploration — our current studies include applying his conceptual framework for the Poincaré sphere so to open discussions about polarization and its applicability within the dynamics of the Fourier transform. Ostensibly all these dynamics, including Fourier A and Fourier B (below) and the Lorentz transformation, are part of Notation #1 and 2.
That first infinitesimal sphere brings with it a huge agenda, but a deeper analysis of period doubling has been blocked by big bang cosmology.
Here we can begin to discern the mathematics and geometries that are the basis of period doublings bifurcations. Of course, we’ll continue looking at sphere stacking. We’re learning how to dance in this sphere of influence. Still entirely clumsy, there is a lot to learn.
In a recent article, I said, “...now included are Mandelbrot’s work on fractals, the Santa Fe Institute and their work on complexity and chaos theory, and Stephen Wolfram on computational irreducibility. In 2006 Ari Lehto refocused his work to explore period doubling at the Planck scale and in 2014 Charles Tresser added insights regarding its universality.”
For a review, go to: https://81018.com/transformation/
https://81018.com/transformation/#9b and https://81018.com/e8/