Topological interpretation of subharmonic mode locking in coupled oscillators with inertia

Abstract

A topological argument is constructed and applied to explain subharmonic mode locking in a system of coupled oscillators with inertia. Via a series of transformations, the system is shown to be described by a classical XY model with periodic bond angles, which is in turn mapped onto a tightbinding particle in a periodic gauge field. It is then revealed that subharmonic quantization of the average phase velocity follows as a manifestation of topological invariance. Ubiquity of multistability and associated hysteresis are also pointed out.PACS numbers: 05.45. Xt, 74.50.+r In a nonlinear oscillator system driven periodically, the competition between the natural frequency and the driving frequency in general leads to either an almost periodic motion or a periodic one, depending on the parameter range [1]. The latter, called mode locking, is characterized by the quantization at rational values of the average phase velocity. In particular subharmonic mode locking, appearing in the presence of the inertia term, results in the devil's staircase structure. One of the well-known examples is the Josephson junction driven by combined direct and alternating currents, with the capacitance playing the role of inertia [2]. Governed by the same equation of motion as a driven pendulum, it displays dc voltage plateaus in the current-voltage characteristics, known as Shapiro steps [3]. Similar voltage quantization has also been observed in arrays of Josephson junctions, yielding integer giant Shapiro steps [4,5] and subharmonic steps [6] according to the absence/presence of capacitive terms. Unfortunately, in spite of the deceptively simple equation of motion, even the single-junction problem has resisted complete analytical solutions, especially, in the presence of the capacitive term, except for the results mainly based on the circle map [7] and on the approximate analysis by means of expansion and averaging [8,9]. Accordingly, such mode locking phenomena in arrays have been demonstrated mostly by numerical simulations. On the other hand, the topological argument, proposed for the system without the capacitive term [10], reveals topological invariance of the system as the physical origin of quantization [11]. As in the case of the quantum Hall effect [12], the topological argument does not provide quantitative information, e.g., on the locking structure. Nevertheless it not only clarifies the nature of quantization but also provides a link between dynamics and statics by interpreting (dynamical) mode locking in terms of (static) topological invariance.In this work, we construct a topological argument for the system with inertia, and apply the idea to Josephsonjunction arrays or systems of coupled oscillators, with attention to the resulting subharmonic locking. For this purpose, we consider an appropriate canonical transformation of the dynamic equations of motion and the corresponding Fokker-Planck equation, the stationary solution of which gives the effective Hamiltonian in the form of a classical XY model with periodic bond...

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