Farey sequence in the appearance of subharmonic Shapiro steps

TL;DR

This paper investigates the appearance of subharmonic Shapiro steps in a driven Frenkel-Kontorova model with deformable potential, revealing Farey sequence patterns in the response function without chaos.

Contribution

It demonstrates the presence of Farey sequence in subharmonic steps of a deformable Frenkel-Kontorova model using Lyapunov exponents, extending understanding of mode-locking phenomena.

Findings

Farey sequence governs the size and appearance of subharmonic steps

No chaos observed in the system despite complex step structure

Farey construction persists in both standard and nonstandard regimes

Abstract

Largest Lyapunov exponent has been examined in the dynamical-mode locking phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with deformable substrate potential. Due to deformation, large fractional and higher order subharmonic steps appear in the response function of the system. Computation of the largest Lyapunov exponent as a way to verify their presence led to the observation of the Farey sequence. In the standard regime, between the large harmonic steps, the appearance of halfinteger and subharmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, though halfinteger steps are larger than harmonic ones, Farey construction is still present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents also shows that there is no chaos in the system.