Devil's staircase and the absence of chaos in the dc- and ac-driven overdamped Frenkel-Kontorova model

Abstract

The devil's staircase structure arising from the complete mode locking of an entirely nonchaotic system, the overdamped dc+ac driven Frenkel-Kontorova model with deformable substrate potential, was observed. Even though no chaos was found, a hierarchical ordering of the Shapiro steps was made possible through the use of a previously introduced continued fraction formula. The absence of chaos, deduced here from Lyapunov exponent analyses, can be attributed to the overdamped character and the Middleton no-passing rule. A comparative analysis of a one-dimensional stack of Josephson junctions confirmed the disappearance of chaos with increasing dissipation. Other common dynamic features were also identified through this comparison. A detailed analysis of the amplitude dependence of the Shapiro steps revealed that only for the case of a purely sinusoidal substrate potential did the relative sizes of the steps follow a Farey sequence. For nonsinusoidal (deformed) potentials, the symmetry of the Stern-Brocot tree, depicting all members of particular Farey sequence, was seen to be increasingly broken, with certain steps being more prominent and their relative sizes not following the Farey rule.

Newton trajectories for the tilted Frenkel–Kontorova model

“…Additionally to the two forces of the FK model, we use an external, linear force in the ansatz [1,2,30,36,[43][44][45][46][47][48][49][50][51][52]. We name the resulting PES an effective PES…”

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

“…The value 1 is the 'trivial' one where every atom has its own well of the on-site potential, 2/3 is used in Ref. [28], 1/2 makes very symmetric structures of the chain [29,30], and the value 0.61803 is the reciprocal golden mean, an often used value to demonstrate the "incommensurability" -especially for the FK model [31]. Compare also Ref.…”

“…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.…”