  We show that the threshold condition for two pump photons to convert into a
pair of the sideband ones in Kerr microresonators with high-quality factors
breaks the pump laser parameter space into a sequence of narrow in frequency
and broad in power Arnold tongues. Instability tongues become a dominant
feature in resonators with the finesse dispersion parameter close to and above
one. As pump power is increased, the tongues expand and cross by forming a line
of cusps, i.e., the threshold of complexity, where more sideband pairs become
unstable. We elaborate theory for the tongues and threshold of complexity, and
report the synchronisation and frequency-domain symmetry breaking effects
inside the tongues.
