2025/04/28 by Artem Alexandrov, Alexandrov, Artem, Alexey Glutsyuk +3 · 1 citation
Computer Science · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Laser-Matter Interactions and Applications #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2504.20181
openalex publication_date 2025/04/28 · openalex created_date 2025/10/05 · openalex updated_date 2026/07/28
In this study, we discuss the Prufer transform that connects the dynamical system on the torus and the Hill equation, which is interpreted as either the equation of motion for the parametric oscillator or the Schrodinger equation with periodic potential. The structure of phase-locking domains in the dynamical system on torus is mapped into the band-gap structure of the Hill equation. For the parametric oscillator, we provide the relation between the non-adiabatic Hannay angle and the Poincare rotation number of the corresponding dynamical system. In terms of quantum mechanics, the integer rotation number is connected to the quantization number via the Milne quantization approach and exact WKB. Using recent results concerning the exact WKB approach in quantum mechanics, we discuss the possible non-perturbative effects in the dynamical systems on the torus and for parametric oscillator. The semiclassical WKB is interpreted in the framework of a slow-fast dynamical system. The link between the classification of the coadjoint Virasoro orbits and the Hill equation yields a classification of the phase-locking domains in the parameter space in terms of the classification of Virasoro orbits. Our picture is supported by numerical simulations for the model of the Josephson junction and Mathieu equation.