Farey sequence in the appearance of subharmonic Shapiro steps

Area of Science:

• Nonlinear dynamics
• Condensed matter physics

Background:

• The Frenkel-Kontorova model describes interacting chains of atoms or particles.
• Dynamical-mode locking and substrate potential deformation are key phenomena in such systems.

Purpose of the Study:

• To investigate the largest Lyapunov exponent in an ac+dc driven dissipative Frenkel-Kontorova model with a deformable substrate potential.
• To analyze the impact of substrate deformation on dynamical-mode locking and system response.

Main Methods:

• Numerical computation of the largest Lyapunov exponent.
• Analysis of the system's response function under varying conditions.
• Examination of the Farey sequence in relation to observed steps.

Main Results:

• Deformation of the substrate potential leads to the emergence of large fractional and higher-order subharmonic steps.
• The Farey sequence was observed in the appearance and relative sizes of subharmonic steps, particularly in the standard regime.
• In the nonstandard regime, half-integer steps dominated harmonic ones, with Farey construction limited to higher-order subharmonics.
• No chaotic behavior was detected in the system, irrespective of substrate potential or deformation.

Conclusions:

• Substrate deformation significantly alters the dynamical-mode locking behavior of the Frenkel-Kontorova model.
• The Farey sequence provides a framework for understanding the complex subharmonic structures observed.
• The system remains non-chaotic even with complex dynamical responses and substrate modifications.