  Log-periodic oscillations have been found to decorate the usual power law
behavior found to describe the approach to a critical point, when the
continuous scale-invariance symmetry is partially broken into a discrete-scale
invariance (DSI) symmetry. We classify the `Weierstrass-type'' solutions of the
renormalization group equation F(x)= g(x)+(1/m)F(g x) into two classes
characterized by the amplitudes A(n) of the power law series expansion. These
two classes are separated by a novel ``critical'' point. Growth processes
(DLA), rupture, earthquake and financial crashes seem to be characterized by
oscillatory or bounded regular microscopic functions g(x) that lead to a slow
power law decay of A(n), giving strong log-periodic amplitudes. In contrast,
the regular function g(x) of statistical physics models with
``ferromagnetic''-type interactions at equibrium involves unbound logarithms of
polynomials of the control variable that lead to a fast exponential decay of
A(n) giving weak log-periodic amplitudes and smoothed observables. These two
classes of behavior can be traced back to the existence or abscence of
``antiferromagnetic'' or ``dipolar''-type interactions which, when present,
make the Green functions non-monotonous oscillatory and favor spatial modulated
patterns.
