Dynamics of phase slips in systems with time-periodic modulation

TL;DR

This paper investigates how periodic frequency modulation affects phase slip dynamics in the Adler equation, revealing resonance phenomena, complex parameter space structures, and the effectiveness of adiabatic approximations across modulation regimes.

Contribution

It provides a detailed analysis of phase slip behavior under time-periodic modulation, combining numerical, asymptotic, and simulation methods to uncover resonance effects and canard trajectories.

Findings

Resonances between modulation period and phase slip generation identified.

Parameter space shows regions with integer and noninteger phase slips.

Adiabatic approximation is effective over a wide range of low frequencies.

Abstract

The Adler equation with time-periodic frequency modulation is studied. A series of resonances between the period of the frequency modulation and the time scale for the generation of a phase slip is identified. The resulting parameter space structure is determined using a combination of numerical continuation, time simulations and asymptotic methods. Regions with an integer number of phase slips per period are separated by regions with noninteger numbers of phase slips, and include canard trajectories that drift along unstable equilibria. Both high and low frequency modulation is considered. An adiabatic description of the low frequency modulation regime is found to be accurate over a large range of modulation periods.