Bifurcation theory and Period Doubling (an emerging document)

By Bruce Camber (Initiated in December 2017)

In 2006 Ari Lehto refocused his work as he explored period doubling at the Planck scale and in 2014 Charles Tresser added insights regarding its universality. Yet, mechanisms to trigger period doubling have not been well understood. Our simple solution: consider sphere stacking to be the doubling mechanism for period doubling bifurcations, particularly at or near the Planck scale.

Henri Poincaré was perhaps the first to provide a detailed description of period doubling bifurcation with his section, maps, and metrics. Mitchell Feigenbaum’s discovery of the precise values for when there is a bifurcation, a ratio of convergence (4.669201609…), was postulated to be the result of the dynamics and symmetries within sphere stacking. In 2008 Ari Lehto opened his exploration of period doubling at the Planck scale and in 2014 Charles Tresser opened explorations about its universality.

Explorations of period doubling: Transformation, August 2019.


References & Resources

Period doubling: Nine key concepts within this section will be further developed. We have barely smudged the surface. The anticipation, perhaps a hope or a dream or fantasy, is that all those disciplines (and all their scholarship) that have been off the grid and in their own silos will eventually discover a place within particular notations or range of notations.

Universal cascades of period doubling bifurcations

Topological Universality and other precursors and asides

“Before metric universality was discovered, combinatorial and topological forms of universality were described for real maps, for instance in a theorem by Sharkovskii (1964) that remained too long mostly unknown [see also Metropolis (1973); Milnor and Thurston (1988), and RG-related ad hoc theories were formulated to understand some of that Gumowski and Mira (1975); Derrida et al., 1978).”

Link chemical bonding and cellular development:

Consider as many things as possible that are in a continuous state of doubling, what do you see?

Perhaps these definitions are too confining:

“…a continuous family fμ, a period-doubling is a “bifurcation” whereby a τ-periodic orbit O0,μ looses its ‘stability as the parameter μ crosses the critical valueμc of μ (we will assume, w.l.o.g. that μ crosses from below), and at which point either:

  • a stable 2τ-periodic orbit emerges (supercritical period doubling); or
  • an unstable 2τ-periodic orbit coalesces with O0,μ and is destroyed (subcritical period doubling).”


In Process: Langlands IV

Enough is enough. It’s over.  The old Big Bang Theory is dead. The one birthed in part by Friedman in 1922, then Lemaître, then Hawking and Guth.