# 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).”

### 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,