  Large blue tilted spectral index axionic isocurvature perturbations can be
produced when the axion sector is far out of equilibrium during inflation
through an initial Peccei-Quinn (PQ) symmetry breaking field displacement along
a nearly flat direction in the effective potential. As a companion to a
previous work, we present analytic formulae for the blue isocurvature spectrum
for the case of the kinetic energy density of the PQ symmetry breaking field
being larger than the quartic power of the final spontaneous PQ symmetry
breaking scale. It corresponds to a regime in which the nonlinearities of the
classical potential become important many times during the formation of the
axion isocurvature quantum perturbations leading to interesting resonant
behavior. One consequence of this nonlinearity-driven resonance is the chaotic
nature of the map that links the underlying Lagrangian parameters to the
isocurvature amplitudes. We point out an accidental duality symmetry between
the perturbation equations and the background field equations that can be used
to understand this. Finally, we present two types of analytic results. The
first relies on a computation utilizing an effective potential wherein fast
time scale fluctuations have been integrated out. The second is grounded in a
functional ansatz, requiring only a limited set of fitting parameters. Both
analytic results should be useful for carrying out forecasts and fits to the
data.
