Bifurcation theory and Period Doubling.

For research  and reference only:

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 value μc 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).”

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 all things that we know are in a continuous state of doubling.  What do you see?

 

In Process: Langlands IV

Enough is enough.

It’s over.

The old Big Bang Theory is dead. Birthed in part by Friedman in 1922,