Multifrequency phase locking and hyperplane structures in particle orientation

Flakelike particles suspended in fluid flows exhibit complex orientational dynamics that are highly sensitive to local shear and vorticity. To investigate the conditions under which their initially isotropic orientation distributions evolve into anisotropic states, we model particle orientation as a nonlinear phase oscillator subjected to multifrequency external forcing along Lagrangian trajectories. By applying Fourier decomposition to time-dependent velocity gradients, we identify resonance conditions and formulate phase-locking phenomena as hyperplanes with finite width in multidimensional frequency space. Numerical simulations confirm the emergence, collision, and persistence of these locking hyperplanes, which correspond structurally to Arnold tongues and devil's staircases known from classical synchronization theory. This framework quantitatively predicts anisotropic orientation alignment and unifies particle aggregation with externally driven synchronization phenomena. Beyond fluid mechanics, the generalized description provides a versatile theoretical basis for multifrequency locking in diverse nonlinear oscillator systems such as optics, thermoacoustics, and optomechanics.