  We consider classically scale-invariant theories with non-minimally coupled
scalar fields, where the Planck mass and the hierarchy of physical scales are
dynamically generated. The classical theories possess a fixed point, where
scale invariance is spontaneously broken. In these theories, however, the
Planck mass becomes unstable in the presence of explicit sources of scale
invariance breaking, such as non-relativistic matter and cosmological constant
terms. We quantify the constraints on such classical models from Big Bang
Nucleosynthesis that lead to an upper bound on the non-minimal coupling and
require trans-Planckian field values. We show that quantum corrections to the
scalar potential can stabilise the fixed point close to the minimum of the
Coleman-Weinberg potential. The time-averaged motion of the evolving fixed
point is strongly suppressed, thus the limits on the evolving gravitational
constant from Big Bang Nucleosynthesis and other measurements do not presently
constrain this class of theories. Field oscillations around the fixed point, if
not damped, contribute to the dark matter density of the Universe.
