Jasmina Tekić

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

Abstract Dynamics of driven dissipative Frenkel–Kontorova model is examined by using largest Lyapunov exponent computational technique. Obtained results show that besides the usual way where behavior of the system in the presence of external forces is studied by analyzing its dynamical response function, the largest Lyapunov exponent analysis can represent a very convenient tool to examine system dynamics. In the dc driven systems, the critical depinning force for particular structure could be estimated by computing the largest Lyapunov exponent. In the dc+ac driven systems, if the substrate potential is the standard sinusoidal one, calculation of the largest Lyapunov exponent offers a more sensitive way to detect the presence of Shapiro steps. When the amplitude of the ac force is varied the behavior of the largest Lyapunov exponent in the pinned regime completely reflects the behavior of Shapiro steps and the critical depinning force, in particular, it represents the mirror image of the amplitude dependence of critical depinning force. This points out an advantage of this technique since by calculating the largest Lyapunov exponent in the pinned regime we can get an insight into the dynamics of the system when driving forces are applied. Additionally, the system is shown to be not chaotic even in the case of incommensurate structures and large amplitudes of external force, which is a consequence of overdampness of the model and the Middleton’s no passing rule.

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

The devils 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.

The friction phenomena in underdamped three layers Frenkel–Kontorova model

Hysteretic phenomenon is studied for an underdampedcommensurate three layers model consisting of two harmonic chains that both move over the substrate. In the depinning process, two critical points and are found. In the pinning process, two critical points and are also found. The dependence of four critical forces on the system parameters is investigated. The analytical result of critical force is obtained which is in good agreement with the numerical one.

Dynamics of the overdamped three-layer model with the incommensurate structure

ABSTRACT An overdamped three-layer model with incommensurate structure consisting of two harmonic chains which both move over the substrate potential is studied in the presence of an external driving force. The force is applied only on the upper harmonic chain, and dynamics of the middle layer is examined. The results show that the dynamical mode-locking appears not only in the upper layer but also in the middle one. The behavior of the middle layer is strongly determined by the amplitude and frequency of the ac force, and the phase diagram which presents pinned, dynamical mode-locking and sliding regimes as a function of system parameters are given. Contrary to the standard case, critical depinning force and steps exhibit anomalous behavior. Particles trajectories have been analyzed which showed very different motion of particles from the one which usually appears in systems under forces.