Analytical fixed point for circular motion

Derive an analytical fixed-point solution for the circular motion in the polar-coordinate formulation of the model of two particles (an active particle with constant self-propulsion force f2 and a passive particle with constant nonreciprocal repulsive force f1) connected by a linear spring and subject to drag η; specifically, obtain closed-form steady values of the reduced variables defining the circular motion without relying on numerical computation.

Background

The paper proposes a minimal model for a pairing-induced motion of an active and a passive particle connected by a linear spring, with the active particle driven along its velocity direction (force f2) and the passive particle repelled from the active particle (force f1), both subject to drag η.

To analyze circular motion, the authors transform the dynamics into polar coordinates, yielding an effectively five-variable dynamical system whose circular motion corresponds to a fixed point. While they numerically compute and confirm the existence and stability of this fixed point under certain conditions, they explicitly state they cannot obtain it analytically, leaving a gap for an analytical derivation.