Nonlinear dynamics of globally coupled oscillators: exact solutions

In the present paper we study analytically the nonlinear dynamics of globally coupled oscillators in the framework of order parameters approach. The main result is the exact analytical solution of the nonlinear system describing the behavior of the oscillators in the orthogonal reduction. We have obtained complete description of all possible phase portraits, the exact expressions for the attracting manifolds including the limit cycle and have derived analytical solutions for arbitrary initial data