  We study systems with a crossover parameter lambda, such as the temperature
T, which has a threshold value lambda* across which the correlation function
changes from exhibiting fixed wavelength (or time period) modulations to
continuously varying modulation lengths (or times). We report on a new
exponent, nuL, characterizing the universal nature of this crossover. These
exponents, similar to standard correlation length exponents, are obtained from
motion of the poles of the momentum (or frequency) space correlation functions
in the complex k-plane (or omega-plane) as the parameter lambda is varied. Near
the crossover, the characteristic modulation wave-vector KR on the variable
modulation length "phase" is related to that on the fixed modulation length
side, q via |KR-q|\propto|T-T*|^{nuL}. We find, in general, that nuL=1/2. In
some special instances, nuL may attain other rational values. We extend this
result to general problems in which the eigenvalue of an operator or a pole
characterizing general response functions may attain a constant real (or
imaginary) part beyond a particular threshold value, lambda*. We discuss
extensions of this result to multiple other arenas. These include the ANNNI
model. By extending our considerations, we comment on relations pertaining not
only to the modulation lengths (or times) but also to the standard correlation
lengths (or times). We introduce the notion of a Josephson timescale. We
comment on the presence of "chaotic" modulations in "soft-spin" and other
systems. These relate to glass type features. We discuss applications to Fermi
systems - with particular application to metal to band insulator transitions,
change of Fermi surface topology, divergent effective masses, Dirac systems,
and topological insulators. Both regular periodic and glassy (and spatially
chaotic behavior) may be found in strongly correlated electronic systems.
