Farey sequence in the appearance of subharmonic Shapiro steps

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.