Researchers developed a data-driven method that uncovers the slow, long-term motion of a simple but nonlinear experimental system: pairs of tiny cubes held in an acoustic trap. The cubes float at a stable height while hinging at a shared edge. By combining careful experiments with a new fitting procedure, the team extracted a compact dynamical model that reproduces how the pair oscillates for hundreds of cycles and explains why those motion patterns are stable.

The experiment uses a standing sound wave in air created by an ultrasonic transducer (frequency 45.650 kHz, wavelength about 7.5 mm). Small salt cubes, 400–700 µm on a side, levitate at the wave node. Two cubes are drawn together by acoustic scattering and form an edge-to-edge contact that acts like a flexible hinge. The authors track two main variables: the vertical position of the cluster’s center of mass (y) and the bending angle of the hinge (θ). They also briefly turn the acoustic field off and back on to drive the system far from equilibrium and observe large oscillations.

For small motions both variables behave like simple springs and oscillate at a single frequency. But after the brief field modulation the researchers observed richer behavior. The vertical motion y remained nearly sinusoidal at one frequency. The hinge angle θ showed two frequency components: a slow component locked to the vertical motion and a fast component at twice that frequency. The phase relationship between y and θ was strongly locked: the cubes settled into one of a few long-lived patterns, or attractors, that persist for hundreds of cycles.

To explain these patterns the authors used a two-stage, data-driven fitting approach. First they fit a flexible polynomial model (up to third order) to recover instantaneous accelerations from the measured time series. That initial fit reproduced short-time accelerations but failed to predict the long-time attractor structure. They then adjusted the model coefficients to match long-time features: the attractor shape, how trajectories converge onto it, and the average oscillation timescale. Repeating the procedure across many experimental runs and keeping only statistically consistent terms produced a minimal nonlinear model.