Farey sequence in the appearance of subharmonic Shapiro steps

Abstract

The 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, the appearance of half-integer and other subharmonic steps between the large harmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, however, the half-integer steps are larger than harmonic ones, and Farey construction is only present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents has also shown that regardless of the substrate potential or deformation, there was no chaos in the system.

Application of largest Lyapunov exponent analysis on the studies of dynamics under external forces

“…Detailed explanation of our approach and numerical procedure can be found in Ref. [27]. In further text, we will refer to the quantity of largest Lyapunov exponent as the Lyapunov exponent (LE) for convenience.…”

Sliding paths for series of Frenkel-Kontorova models – a contribution to the concept of 1D-superlubricity

“…The new term with the force F is often named dc driving (dc: direct current) [2,58]. In many applications it is further subdivided in the dc-part, F dc , and an ac-part (alternating current) [30,32]. It is F ac cos(c t) with a constant c and a time variable t. The force tilts the former on-site potential with the incline F .…”

“…The two edge-driven forces are given, but the internal movement of the chain is a black box. This direction is sometimes 'quite better' for NTs than others, like the only push direction, (1,0,...,0) T , or the only-pull direction, (0,...,0,1) T , or an equal force [30,32] to all particles, the direction (1,1,...,1,1) T , making a tilted washboard potential. 'Quite better' here means the property of the corresponding NT to find a minimum energy pathway (MEP) through the PES mountains, or at least a similar low energy path (LEP).…”

Vibrational resonance in an oscillator with an asymmetrical deformable potential

“…Remoissenet and Peyrard [47,48] proposed deformable models of nonlinear systems with deformable periodic substrate potential V RP (x, r), where the parameter, r is the deformable po-tential parameter which determines the shape of the potential. The potential V RP (x, r), now often known as the Remoissenet-Peyrard-potential, plays a significant role in one-dimensional atomic chains and has become a subject of diverse research focus [49][50][51][52][53][54][55][56][57][58][59][60][61]. Earlier studies, for instance, Nana, et al [49] used Melnikov theory to predict the onset of chaotic behaviour of a particle in an asymmetric doubly-periodic potential.…”

“…Mali et al [53] observed the appearance of large subharmonic Shapiro steps due to deviations from the sinusoidal potential. More recently, stochastic resonance [54], Farey sequences and Shapiro steps [55], anomalous transport and diffusion phenomena [56][57][58], current reversals [59], and jump diffusion [60], as well as Devil's staircases [61], have all been investigated. In general, the dynamical and statistical properties are strongly dependent on the variation of the biharmonic parameter, the shape parameter and the phase-lag.…”