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We show that the threshold condition for two pump photons to convert into apair of the sideband ones in Kerr microresonators with high-quality factorsbreaks the pump laser parameter space into a sequence of narrow in frequencyand broad in power Arnold tongues. Instability tongues become a dominantfeature in resonators with the finesse dispersion parameter close to and aboveone. As pump power is increased, the tongues expand and cross by forming a lineof cusps, i.e., the threshold of complexity, where more sideband pairs becomeunstable. We elaborate theory for the tongues and threshold of complexity, andreport the synchronisation and frequency-domain symmetry breaking effectsinside the tongues.

Threshold of complexity and Arnold tongues in Kerr ring microresonators