Phase Locking and Discrete Mode Selection in Closed Circulation Loops

Closed circulation loops with finite propagation time constitute a broad and generic class of delayed feedback systems encountered across fluid dynamics, optics, electronics, and wave physics. In this work we show that such loops admit discrete, dynamically selected phase--locked states even when governed by fully continuous classical dynamics. Using a minimal delayed phase oscillator as a canonical model, we demonstrate that finite circulation time enforces multistable phase--locking conditions indexed by an integer winding number. These phase--locked solutions form a discrete family of circulation modes and provide a purely classical route to mode discreteness in circulating systems, independent of topology, microscopic quantization rules, or particle-based assumptions. The analysis establishes phase locking in long--delay loops as a universal and robust mechanism for the emergence of discrete circulation modes.