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Visualization of oscillatory electron dynamics on the surface of liquid heliumWe investigate the time traces of currents induced in five segmented electrodes by the motion of electrons on the surface of liquid $^{4}\mathrm{He}$ at $\ensuremath{\sim}0.3$ K, that are placed in a perpendicular magnetic field and exposed to microwave radiation. Nonlinear dynamics methods are utilized to explore the characteristic features of the current oscillations in the electrodes for different electron densities and pressing voltages. The wavelet phase coherence and phase shift are calculated to obtain the coherence relationships between the signals in all five electrodes as functions of the pressing voltage. Coupling analysis of the ridge-extracted instantaneous frequencies revealed the directions of motion of electrons inside the cell and provided evidence of strong phase coupling at a pressing voltage of 4.20 V. These classical methods reveal that the motion is oscillatory, with a varying frequency subject to a constant frequency modulation. High harmonics due to nonlinearity arise at higher frequencies where the resonance condition is satisfied at a pressing voltage of 4.20 V for low-electron density. Our approach provides a platform for investigating these phenomena analytically. We show that slow gravity waves on the helium surface modulate the electronic oscillatory behavior and illustrate that the model in fact produces three-dimensional dynamics. Motion of electrons on the surface of liquid helium is shown to be a paradigmatic example of a chronotaxic system, i.e., a system that undergoes continuous perturbation but is nonetheless capable of maintaining its stability.
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