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The Ott--Antonsen ansatz is a powerful tool to extract the behaviors ofcoupled phase oscillators, but it imposes a strong restriction on the initialcondition. Herein, a systematic extension of the Ott--Antonsen ansatz isproposed to relax the restriction, enabling the systematic approximation of thebehavior of a globally coupled phase oscillator system with an arbitraryinitial condition. The proposed method is validated on the Kuramoto--Sakaguchimodel of identical phase oscillators. The method yields cluster andchimera-like solutions that are not obtained by the conventional ansatz.

physical review. e

Diversity of dynamical behaviors due to initial conditions: exact  results with extended Ott--Antonsen ansatz for identical Kuramoto--Sakaguchi  phase oscillators