My question is under what conditions Buddha arises?
There's no set patterns. For example Buddha Gotama came during a relatively uneventful period where human life span was within normal range. However, for the future Buddha Metteyya to arise, per DN 26, life on earth must undergo some very drastic "downswing", and then some recovery, and finally thru some great upswing before He'll appear.
To be buddha merely means to be awake: to open your eyes and see the true nature of reality. That happens more often than anyone realizes, to people one might never hear about or expect.
To be a teacher — one with open eyes who goes beyond the karma of others as well as his own — is rare. I suspect that the world has to be locked in dukkha, a generalized state of dissolute and decayed dissatisfaction, in order to effectively pull people to the path without pre-existing structures. No one tears down an old temple to build a new unless the old one is close to rubble and ruin. But that's more an intuition than anything else.